Methods of spectral analysis in mathematical physics: Conference, OTAMP 2006, Lund, Sweden

Operator Theory: Advances and Applications Vol. 186 Editor: I. Gohberg Editorial Office: School of Mathematical Scienc...

Author: Jan Janas | Pavel Kurasov | Ari Laptev | Sergei Naboko | Günter Stolz

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Operator Theory: Advances and Applications Vol. 186

Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv Israel

Editorial Board: D. Alpay (Beer Sheva, Israel) J. Arazy (Haifa, Israel) A. Atzmon (Tel Aviv, Israel) J.A. Ball (Blacksburg, VA, USA) H. Bart (Rotterdam, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) H. Bercovici (Bloomington, IN, USA) A. Böttcher (Chemnitz, Germany) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) M. Demuth (Clausthal-Zellerfeld, Germany) A. Dijksma (Groningen, The Netherlands) R. G. Douglas (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) A. Ferreira dos Santos (Lisboa, Portugal) A.E. Frazho (West Lafayette, IN, USA) P.A. Fuhrmann (Beer Sheva, Israel) B. Gramsch (Mainz, Germany) H.G. Kaper (Argonne, IL, USA) S.T. Kuroda (Tokyo, Japan) L.E. Lerer (Haifa, Israel) B. Mityagin (Columbus, OH, USA)

V. Olshevski (Storrs, CT, USA) M. Putinar (Santa Barbara, CA, USA) A.C.M. Ran (Amsterdam, The Netherlands) L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) I.M. Spitkovsky (Williamsburg, VA, USA) S. Treil (Providence, RI, USA) C. Tretter (Bern, Switzerland) H. Upmeier (Marburg, Germany) N. Vasilevski (Mexico, D.F., Mexico) S. Verduyn Lunel (Leiden, The Netherlands) D. Voiculescu (Berkeley, CA, USA) D. Xia (Nashville, TN, USA) D. Yafaev (Rennes, France)

Honorary and Advisory Editorial Board: L.A. Coburn (Buffalo, NY, USA) H. Dym (Rehovot, Israel) C. Foias (College Station, TX, USA) J.W. Helton (San Diego, CA, USA) T. Kailath (Stanford, CA, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P. Lancaster (Calgary, AB, Canada) H. Langer (Vienna, Austria) P.D. Lax (New York, NY, USA) D. Sarason (Berkeley, CA, USA) B. Silbermann (Chemnitz, Germany) H. Widom (Santa Cruz, CA, USA)

Methods of Spectral Analysis in Mathematical Physics Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006, Lund, Sweden

Jan Janas Pavel Kurasov Ari Laptev Sergei Naboko Günter Stolz Editors

Birkhäuser Basel · Boston · Berlin

Editors: Jan Janas Institut of Mathematics Polish Academy of Sciences ul. Sw. Tomasza 30/7 31-027 Krakow, Poland e-mail: [email protected] Pavel Kurasov Department of Mathematics Lund Institute of Technology 221 00 Lund, Sweden e-mail: [email protected] Sergei Naboko Department of Mathematical Physics St. Petersburg State University Ulyanovskaya 1 198904 St. Petersburg, Russia e-mail: [email protected]

Ari Laptev Department of Mathematics South Kensington Campus Imperial College London SW7 2AZ London, UK e-mail:[email protected] and Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden e-mail: [email protected]

Günter Stolz Department of Mathematics University of Alabama 452 Campbell Hall Birmingham, AL 35294-1170, USA e-mail: [email protected]

2000 Mathematical Subject Classification: 05C25, 34A55, 34B, 34L40, 35J10, 35K, 35P, 37K, 42C05, 47, 49N45, 81Q, 81T99, 82B43 Library of Congress Control Number: 2008935517 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8754-9 Birkhäuser Verlag AG, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8754-9 e-ISBN 978-3-7643-8755-6 987654321 www.birkhauser.ch

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

T. Antunović and I. Veselić Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

B.M. Brown and R. Weikard On Inverse Problems for Finite Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

E.V. Cheremnikh Wave Operators for Nonlocal Sturm-Liouville Operators with Trivial Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

M. Combescure Semiclassical Results for Ideal Fermion Systems. A Review . . . . . . . . . . .

69

J. Derezi´ nski Quadratic Hamiltonians and Their Renormalization . . . . . . . . . . . . . . . . . .

89

J. Dombrowski Jacobi Matrices: Eigenvalues and Spectral Gaps . . . . . . . . . . . . . . . . . . . . .

103

R.L. Frank and A.M. Hansson Eigenvalue Estimates for the Aharonov-Bohm Operator in a Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 F. Gesztesy, H. Holden, J. Michor and G. Teschl The Ablowitz–Ladik Hierarchy Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 F. Gesztesy, M. Mitrea and M. Zinchenko On Dirichlet-to-Neumann Maps and Some Applications to Modified Fredholm Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191

D.J. Gilbert, B.J. Harris and S.M. Riehl Higher Derivatives of Spectral Functions Associated with One-dimensional Schr¨ odinger Operators . . . . . . . . . . . . . . . . . . . . . . . .

217

vi

Contents

P.-A. Ivert On the Boundary Value Problem for p-parabolic Equations . . . . . . . . . .

229

M. Kadowaki, H. Nakazawa and K. Watanabe On the Rank One Dissipative Operator and the Parseval Formula . . . . 241 Y. Karpeshina and Y.-R. Lee On the Schr¨ odinger Operator with Limit-periodic Potential in Dimension Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 A.V. Kiselev Similarity Problem for Non-self-adjoint Extensions of Symmetric Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

S. Kupin, F. Peherstorfer, A. Volberg and P. Yuditskii Inverse Scattering Problem for a Special Class of Canonical Systems and Non-linear Fourier Integral. Part I: Asymptotics of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285

A.B. Mikhailova and B.S. Pavlov Remark on the Compensation of Singularities in Krein’s Formula . . . .

325

A. Osipov On Some Properties of Infinite-dimensional Elliptic Coordinates . . . . .

339

F. Peherstorfer and P. Yuditskii Finite Difference Operators with a Finite-Band Spectrum . . . . . . . . . . . . 347 C. Svensson, S. Silvestrov and M. de Jeu Connections Between Dynamical Systems and Crossed Products of Banach Algebras by Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391

F. Truc Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the Area . . . . . . . . . . . . . . . . . 403 I. Wood The Ornstein-Uhlenbeck Semigroup in Bounded and Exterior Lipschitz Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Introduction This volume contains mainly the lectures delivered by the participants of the International Conference: Operator Theory, Analysis and Mathematical Physics – OTAMP2006, held in Lund. As in the previous conferences of the OTAMP series, the main lectures presented an overview of current research which uses operator methods in analysis and mathematical physics. The topics of the Proceedings belong to various different fields of mathematical physics. Among others (but there is much more in this volume) the following subjects are presented: inverse spectral and scattering problems on graphs, review articles on quadratic Hamiltonians and Born-Oppenheimer approximations, recursive construction of the Ablowitz-Ladik Hierarchy, spectral properties of finite difference and one (or two) dimensional Schr¨ odinger operators, eigenvalues and their estimates for Jacobi matrices and Aharonov-Bohm operator. Most papers of the volume contain original material and were refereed by acknowledged experts. The Editors thank all the referees whose job helped to improve the clarity of the collected material. The Organizing Committee of the conference also thanks all session organizers for the interesting choice of the scientific programme and to all participants who delivered fine lectures. We greatly appreciate financial support of the ESF programme SPECT, without whose financial support the OTAMP2006 would never been so well organized. Special thanks go to other agencies supported the conference financially: Vetenskapsr˚ adet, Wenner-Gren Foundation, as well as to Lund University and to the Institute of Mathematics of the Polish Academy of Sciences. Last but not least, we thank the Editorial Board and especially Professor I. Gohberg for including this volume (as the previous ones of OTAMP meetings) into the series Operator Theory: Advances and Applications and to Birkh¨ auserVerlag for help in preparation of the volume. Birmingham-Krakow-Lund St. Petersburg-Stockholm

April 2008 The Editors

Operator Theory: Advances and Applications, Vol. 186, 1–29 c 2008 Birkh¨ auser Verlag Basel/Switzerland

Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs Tonći Antunović and Ivan Veselić Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem. Mathematics Subject Classification (2000). 05C25, 82B43, 05C80, 37A30, 35P15. Keywords. Amenable groups, Cayley graphs, random graphs, percolation, random operators, spectral graph theory, phase transition.

1. Introduction The results on which we report rely on tools and ideas from various mathematical fields. In the introduction we sketch the role they play in our study of percolation Hamiltonians. 1.1. Integrated density of states For a large class of random operators which are ergodic with respect to a group of translations on the configuration space an integrated density of states (or spectral density function) can be defined using an exhaustion of the whole space by subsets of finite volume. Since this fact relies on an ergodic theorem, it is not surprising that the underlying group needs to be amenable. Note that periodic operators are a special class of ergodic ones and thus also posses a well-defined integrated density of states (in the following abbreviated as IDS). For ergodic random operators the spectrum is deterministic, i.e., for any two realizations of the operator the spectrum (as a set) coincides almost surely. The same statement holds for the

2

T. Antunović and I. Veselić

measure theoretic components of the spectrum. However, there are other spectral quantities, like eigenvalues or eigenvectors, which are highly dependent on the randomness. For random Schr¨ odinger type operators on L2 (Rd ) and ℓ2 (Zd ) all these fundamental results can be found, e.g., in the monographs [17, 94]. For the underlying original results of Pastur and Shubin for random, respectively almostperiodic operators see, e.g., [91, 93] and [102, 103]. Once the existence of the IDS is established, it is natural to ask whether, for specific models, one can describe some of its characteristic features in more detail. Among them the most prominent are: the continuity and discontinuity properties of the IDS, and the asymptotic behaviour near the spectral boundaries. The interest in these questions has a motivation stemming from the quantum theory of solids. A good understanding of the features of the IDS is an important step towards the determination of the spectral types of the considered Hamilton operators. Of all the measure theoretic spectral types of random operators the pure point part is so far understood best. In particular, for certain random Hamiltonians it is established that there exists an energy interval with dense eigenvalues and no continuous spectral component, a phenomenon called localisation. For detailed expositions of this subject see, for instance, [3, 5, 37, 101]. All the proofs of localisation known so far use as an essential ingredient estimates on the IDS or some closely related quantity. Expository accounts of the IDS of various types of random Hamiltonians can be found among others in [67], [53] or [113]. 1.2. Lifshitz asymptotics The IDS of lower bounded periodic operators in Euclidean space exhibits typically a polynomial behaviour at the minimum of the spectrum. This is for instance the odinger operators case for the discrete Laplace operator on ℓ2 (Zd ) and for Schr¨ on L2 (Rd ) with a periodic potential. Analogous results can be proven for uniformly elliptic divergence type operators with periodic coefficients using [51] and for quantum waveguides using, e.g., [15] or [60]. In the mentioned cases the low energy behaviour of the IDS is characterised by the so-called van Hove singularity. This means that the IDS N (·) behaves for 0 < E ≪ 1 asymptotically like N (E) ∼ E d/2 , i.e., polynomially with the van Hove exponent equal to d/2. A random perturbation of the Hamilton operator changes drastically the low energy asymptotics. For many random models in d-dimensional Euclidean space it has been proven that the asymptotic behaviour of the IDS is exponential in the sense that N (E) ∼ exp(− const(E − E0 )−d/2 ). Here E0 denotes the minimum of the spectrum of the random operator, which is by ergodicity independent of the realization almost surely. The exponential behaviour of the IDS has been first deduced on physical grounds by I.M. Lifˇsic in [69–71] and is accordingly called Lifshitz asymptotics or Lifshitz tail. The most precise bounds of this type have been obtained for random Schr¨ odinger operators with a potential generated by impurities which are distributed randomly in space according to a Poisson process, see, e.g., [35, 59, 86, 92, 105]. Similar results hold for a discrete relative of this operator, namely the Anderson model on ℓ2 (Zd ), see, e.g., [7, 16, 36, 57, 79, 109]. The

Spectral Asymptotics of Percolation Cayley Graphs

3

reason why these models are amenable to a very precise analysis is the applicability of Brownian motion, respectively random walk techniques and Feynman-Kac functionals. For several other types of random operators the weaker form lim

EցE0

log | log(N (E) − N (E0 ))| d = | log(E − E0 )| 2

(1)

of the exponential law has been established using rather simple estimates based on inequalities by Thirring [107] and Temple [106]. In particular, this method can be applied to show that the asymptotics (1) holds for the Anderson model on ℓ2 (Zd ) and its continuum counterpart, the alloy type model on L2 (Rd ). This strategy of proof was pursued in the 1980s in several papers [52, 55, 80–82, 98, 99] by Kirsch, Martinelli, Simon and Mezincescu. These two models will serve as a point of reference in the present note, since they are closely related to percolation Hamiltonians. Moreover, the methods used to study the low-energy asymptotics of percolation models are, at least in part, parallel to those of the last mentioned series of papers. The main difference is that one needs to replace the Temple or Thirring bound by some inequality from spectral geometry. We will call the righthand side of (1) the Lifshitz exponent. Let us relate the asymptotic behaviour (1) to two spectral features mentioned already in §1.1, namely spectral localisation and continuity of the IDS. If (1) holds then the spectral edge E0 is called a fluctuation boundary. This term stems from the fact, that if the system is restricted to a large, finite volume, eigenvalues close to E0 correspond to very particular and rare realizations of the randomness. Thus the spectral edge on finite volumes is highly sensitive to fluctuations of the random configurations. This feature is closely related to the phenomenon of localisation. Indeed, for many models existence of pure point spectrum has been proven precisely in the energy regimes where the density of states is very sparse, see, e.g., the characterisation given in [38]. The Lifshitz asymptotics implies that the IDS is extremely thin near the bottom of the spectrum, thus this energy region is a typical candidate for pure point spectrum. One expects that the IDS is continuous at the fluctuation boundaries. Indeed, the Lifshitz asymptotics implies continuity at the minimum E0 of the spectrum and moreover that the limes superior and inferior of difference quotients of any order vanish at E0 . This is in remarkable contrast to the dense set of discontinuities of the IDS exhibited by many percolation Hamiltonians, see, e.g., [18, 27, 112]. 1.3. Percolation Hamiltonians Hamiltonians on site percolation subgraphs of the lattice were introduced by P.G. de Gennes, P. Lafore and J. Millot in [26, 27] as quantum mechanical models for binary alloys. The resulting random operator bears a strong similarity to the Anderson model considered by P.W. Anderson in [6]. The difference is that in the site percolation Hamiltonian of [27] the random potential may assume only two values, namely zero and plus infinity. The lattice sites where the potential equals infinity are deleted from the graph, and thus the Hamiltonian is restricted to the

4


space of vertices where the potential vanishes. The resulting Hamiltonian may be understood as a single band approximation of an Anderson model whose potential is a family of Bernoulli distributed random variables. More precisely, if one equips the Bernoulli-Anderson potential with a coupling constant and lets this one tend to infinity, the corresponding quadratic form converges to the one of the quantum percolation model, cf. Remarks 11 and 12. This limit is in the physical literature, cf., e.g., [50], sometimes understood as the strong scattering limit of the tight binding model. It may be possible to estimate efficiently the convergence of the associated IDS using the techniques and estimates introduced in [64]. A series of papers in theoretical [18, 50] and computational physics, e.g., [44– 47, 97], analysed the spectral features of the quantum percolation model. A particular point of interest in the numerical studies was to compare the spectral localisation of percolation Hamiltonians with the one of the Anderson model. More recently there has been interest for such models in the mathematics community. The paper [16], which studies the intermittency behaviour of the parabolic Anderson model, establishes also Lifshitz tails for the quantum site percolation model. Given the above-mentioned relation between the Anderson and the quantum percolation Hamiltonian, it is not surprising that a detailed analysis of the former gives also results about the latter. Some related properties of random walks on percolation graphs have been analysed rigorously already in [7]. The existence of the IDS for a rather general class of site percolation Hamiltonians on graphs with a quasi transitive, free1 amenable group action was established in [111]. This result relies on Lindenstrauss’ pointwise ergodic theorem [72] for locally compact second countable amenable groups. Likewise, the non-randomness of the spectrum and its components is valid also in this general setting, see [65, 112]. These results hold in particular for the Anderson model and periodic Laplacians on such graphs. In fact, these features of the percolation Hamiltonians and their proofs are quite analogous to those of the Anderson model on ℓ2 (Zd ). However, there are some distinct features of the quantum percolation model which distinguish it sharply from random Hamiltonians on the full lattice. The latter have a continuous IDS [24, 28], while percolation Hamiltonians have a dense set of discontinuities, which survives even if one restricts the operator to infinite percolation clusters [18, 50, 112]. In [54] Kirsch & M¨ uller analysed basic spectral features of bond percolation Laplacians on the lattice and moreover carried out a thorough study of the low-lying spectrum of these operators in the non-percolating regime. These results were complemented by M¨ uller & Stollmann in a paper [83] where the percolating regime is studied. The present note continues the analysis [112] of site percolation Hamiltonians on general graphs and at the same time extends the results of [54] to bond percolation models on amenable Cayley graphs. In particular, part of our proofs relies on ideas introduced in [54] and extends them to a more general geometric setting. 1 In

the references [111, 112] is was forgotten to spell out the assumption that the group acts freely on the graph.


5

It is not surprising that the analysis of (certain) spectral properties of percolation Hamiltonians relies on the proper understanding of the underlying ‘classical’ percolation problem. An exposition of this theory applied to independent bond percolation on the lattice can be found in Grimmett’s book [41]. Some other aspects are covered in the monograph [49] by Kesten. In the more recent literature quite a body of work is devoted to percolation processes on more general graphs than Zd , see for instance [13, 14, 76, 96]. Still, much more is known for percolation on lattices than on general graphs. For the purposes of the results presented in this paper, we extended the theorems in [2, 4, 77] on the sharpness of the phase transition and the exponential decay of the cluster size in the subcritical regime to quasi transitive graphs, see [8]. Since in the subcritical regime no infinite percolation cluster exists almost surely, the entire spectrum of the corresponding Hamiltonian consists of eigenvalues. In the supercritical regime an infinite cluster exists and thus the Hamiltonian may have continuous spectrum. It is however conjectured [97] that for values of the percolation parameter just above the critical point the spectrum will still have no continuous component, although an infinite cluster exists. Thus there may be a second ‘quantum’ critical value of the percolation parameter, strictly larger than the classical critical value. Let us note that there is no rigorous proof of Anderson localisation for percolation Hamiltonians (in the percolating regime). The reason is that this model has Bernoulli distributed randomness, as it is the case for the Bernoulli-Anderson Hamiltonian. Due to the singular nature of the distribution of the randomness the known proofs of localisation do not apply. 1.4. Spectral graph theory and geometric L2 -invariants To make sense of the term ‘low energy asymptotics’ one has to know where the minimum of the spectrum lies. In the case of Cayley graphs of amenable groups it is known from Kesten’s theorem [48] that the bottom of the spectrum of the Laplacian is equal to zero. If the graph under consideration happens to be bipartite the spectrum of the adjacency operator is symmetric with respect to the origin, see, e.g., [23, 98]. This allows one to translate results about the lower spectral boundary of adjacency and Laplace operators to statements on the upper boundary. The low energy behaviour of the IDS is determined by the first non-zero eigenvalue of the Hamiltonian on percolation clusters with ‘optimal shape’. The minimizing configurations are determined by inequalities from spectral graph theory like the isoperimetric, the Cheeger and the Faber & Krahn inequality, cf. [19–22]. They relate the lowest eigenvalue on a finite subgraph to its volume and the volume growth behaviour of the Cayley graph. The growth of balls on Cayley graphs, in turn, can be classified using a result of Bass [12] and Gromov’s theorem [42]. We use an improved version of the latter due to van den Dries & Wilkie [108]. Let us mention that even if one is interested in purely geometric properties of the automorphism group of a graph, like amenability or unimodularity, percolation may be a tool of choice, see for instance [13]. There several equivalences are established,

6


each connecting a geometric property of a graph with a probabilistic property of an associated percolation process. For several types of Hamiltonians on the lattice the Lifshitz exponent of the random operator equals the exponent of the van Hove singularity of its periodic counterpart. It turns out that the same holds for certain percolation models on Cayley graphs. To formulate this more properly one needs to characterise the high energy behaviour of the adjacency operator (resp. low energy behaviour of the Laplacian) on the full Cayley graph. This can be done by relating it to known results on random walks on groups, cf. [115], or on geometric L2 -invariants, cf., e.g., [75]. In fact, it turns out that the van Hove exponent of a Cayley graph equals the first Novikov-Shubin invariant of the Laplacian on the graph. These invariants have been introduced in [87, 88] and studied, e.g., in [43]. For various analogies between geometric L2 -invariants and properties of the IDS see the workshop report [31]. The above-mentioned equality of the Lifshitz and the van Hove exponent is encountered also in other contexts, see for instance Klopp’s analysis [56] of Lifshitz tails at internal spectral gap edges.

1.5. The main result Let us loosely state the main result of this note. Consider a Cayley graph of a finitely generated amenable group. Assume that the volume of balls of radius n in the graph behaves like nd with the convention that d = ∞ corresponds to superpolynomial growth. Each deleted site (respectively bond) induces a new boundary in the graph, at which we may impose a certain type of boundary condition giving rise to different Laplace operators. More precisely, since we are dealing with bounded operators, the boundary condition is not a restriction on the domain of the operator, but rather an additional boundary term. As in [54] we consider Dirichlet, Neumann, and adjacency (or pseudo-Dirichlet) percolation Laplacians. For the adjacency and Dirichlet Laplacian the low energy asymptotics of the IDS is given by lim

Eց0

log | log N (E)| d = . | log E| 2

(2)

Thus we have a Lifshitz type behaviour and zero is a fluctuation boundary of the spectrum. The Lifshitz exponent coincides with the van Hove exponent of the underlying full Cayley graph. For the Neumann Laplacian the low energy asymptotics of the IDS is given by lim

Eց0

log | log(N (E) − N (0))| 1 = | log E| 2

(3)

where N (0) is a non-zero value corresponding to the number of open clusters per vertex in the percolation graph.


7

2. Model and results We present several results which we have obtained for the IDS of Hamiltonians on full Cayley graphs and on graphs diluted by a percolation process. For the analysis of the latter operators it was necessary to establish certain properties of the ‘classical percolation model’ on Cayley graphs, which are detailed below. 2.1. The Laplace Hamiltonian on a full Cayley graph We consider Cayley graphs of finitely generated, amenable groups and the corresponding Laplace operator. Its IDS exhibits a van Hove asymptotics at the lower spectral edge whose exponent is determined by the volume growth behaviour of the group. To formulate this more precisely we explain the geometric setting in detail. Let Γ be a finitely generated group. Each choice of a finite, symmetric set of generators S not containing the unit element ι of Γ gives rise to Cayley graph. Note that this graph is regular (i.e., all vertices have the same degree) with degree equal to the number of elements in S and that the vertex set of G can be identified with Γ. Using the distance function on the Cayley graph we define the ball B(n) of radius n around the unit element ι in Γ and set V (n) := |B(n)|. For a positive integer d we use the notation V (n) ∼ nd to signify that there exist constants 0 < a, b < ∞ such that a nd ≤ V (n) ≤ b nd . From now on we tacitly assume that all considered groups are finitely generated. The growth of all such groups can be classified using deep results of Bass [12], Gromov [42] and van den Dries & Wilkie [108]. Theorem 1. Let G be a Cayley graph of a finitely generated group. Then exactly one of the following is true: (a) G has polynomial growth, i.e., V (n) ∼ nd holds for some d ∈ N, (b) G has superpolynomial growth, i.e., for every d ∈ N and every b ∈ R there exists only finitely many integers n such that V (n) ≤ b nd . The growth behaviour (in particular the exponent d) is independent of the chosen set of generators S. More precisely, the theorem follows from the following results. Bass has shown that every nilpotent group satisfies V (n) ∼ nd for some d ∈ N, sharpening earlier upper and lower bounds of Wolf [116]. Gromov’s result in [42] is that every group of polynomially bounded volume growth is virtually nilpotent. Finally, van den Dries & Wilkie have shown that the conclusion of Gromov’s theorem still holds, if one requires the polynomial bound V (nk ) ≤ b ndk merely along a strictly increasing sequence (nk )k∈N of integers nk ∈ N. We note that Pansu [90] has shown that c := limn→∞ V (n)/nd exists. Let us now define the Laplacian on G. For future reference we consider a more general situation than needed at this stage. Let G = (V, E) be a connected regular graph of degree k with vertex set V and edge set E, and G′ = (V ′ , E ′ ) an arbitrary subgraph of G. Note that G′ is in general not regular. We denote the

8


degree of the vertex x ∈ V ′ in G′ by dG′ (x). If two vertices x, y ∈ V ′ are adjacent in the subgraph G′ we write y ∼G′ x. Definition 2. For G and G′ as above we define the following operators on ℓ2 (V ′ ). (a) The identity operator on G′ is denoted by IdV ′ . If there is no danger of confusion we drop the subscript V ′ . (b) The degree operator of G′ acts on ϕ ∈ ℓ2 (V ′ ) according to [D(G′ )ϕ](x) := dG′ (x)ϕ(x).

(c) The adjacency operator on G′ is defined as ϕ(y). [A(G′ )ϕ](x) := y∈V ′ y∼ ′ x G

(d) The adjacency Laplacian on G′ is defined as ∆A (G′ ) := k IdV ′ −A(G′ ).

(e) In the special case G′ = G, the Laplacian or free Hamiltonian on G is defined as ∆(G) := k IdV −A(G). (Thus it coincides with the adjacency Laplacian on G.)

There are several different names used for the adjacency Laplacian ∆A (G′ ) in the literature. Our terminology is motivated by the fact that up to a multiplicative and an additive constant it is equal to the adjacency operator on the subgraph G′ . This will be different for operators with an additional Dirichlet or Neumann boundary term introduced in Subsection 2.3. For induced subgraphs G′ of G one can consider the restriction of the Laplacian which is defined as ∆P (G′ ) := PV ′ ∆(G)PV∗ ′ . Here PV ′ : ℓ2 (V ) → ℓ2 (V ′ ), PV ′ ϕ(x) := χV ′ (x)ϕ(x) denotes the orthogonal projection on V ′ . It turns out that ∆P (G′ ) = k IdV ′ −A(G′ ) = ∆A (G′ ). The IDS of the free Hamiltonian on a Cayley graph G of an amenable group Γ can be defined as N0 (E) := χ]−∞,E] (∆(G))δι , δι . Here the function δι has value 1 at ι and 0 everywhere else. It is possible to construct N0 via an exhaustion procedure, see, e.g., [32, 33, 73]. The next Theorem characterises the asymptotic behaviour of the IDS at the spectral bottom. For groups of polynomial growth it exhibits a van Hove singularity. For groups of superpolynomial growth one encounters a type of generalised van Hove asymptotics with the van Hove exponent equal to infinity. Theorem 3. Let Γ be a finitely generated, amenable group, ∆(G) the Laplace operator on a Cayley graph of Γ and N0 the associated IDS. If Γ has polynomial growth of order d then lim

Eց0

log N0 (E) d = . log E 2

(4)


9

and if Γ has superpolynomial growth then lim

Eց0

log N0 (E) = ∞. log E

(5)

2.2. Classical percolation Next we briefly introduce percolation on graphs. More precisely, we will consider independent site percolation, as well as independent bond percolation on quasi transitive graphs. These are graphs whose vertex set decomposes into finitely many equivalence classes under the action of the automorphism group. Such graphs have uniformly bounded vertex degree. If there is only one orbit, the graph is called transitive. Cayley graphs are particular examples of transitive graphs. Let G = (V, E) be an infinite, connected, quasi transitive graph and p some fixed real number between 0 and 1. For every site x ∈ V we say it is open with probability p and closed with probability 1 − p independently of all other sites. This is the most simple site percolation model. More formally, we consider for every vertex x the probability space Ωx := {0, 1}, with the sigma algebra consisting of the power set P(Ωx ) of Ωx and the probability measure Px on (Ωx , P(Ωx )) defined space (ΩV , FV , PV ) associated by Px (0) = 1 − p and Px (1) = p. The probability to site percolation is defined as the product x∈V (Ωx , P(Ωx ), Px ). The coordinate map ΩV ∋ ω → ωx defines the site percolation process ΩV × V → {0, 1}. We call V (ω) := {x ∈ V | ωx = 1} the set of open or active sites. The induced subgraph of G with vertex set V (ω) is denoted by G(ω) and called the percolation subgraph in the configuration ω. The connected components of G(ω) are called clusters. The bond percolation process is defined completely analogously. The bond percolation probability space is (ΩE , FE , PE ) := e∈E (Ωe , P(Ωe ), Pe ), where for every edge e ∈ E we have Ωe := {0, 1}, with power set P(Ωe ) and probability measure Pe defined by Pe (0) = 1 − p and Pe (1) = p. For a given configuration ω ∈ ΩE we define the percolation subgraph G(ω) as the graph whose edge set E(ω) is the set of all e ∈ E with ωe = 1 and whose vertex set V (ω) consist of all vertices in V which are incident to an element of E(ω). Connected components of G(ω) are called clusters. We will denote the expectation with respect to either PV or PE by E{. . . }. The most basic result in percolation theory is that for both the site and the bond model there exists a critical parameter 0 < pc ≤ 1 such that the following statement holds: • if p < pc there is no infinite cluster almost surely (subcritical phase), • if p > pc there is an infinite cluster almost surely (supercritical phase).

Of course, the value of pc depends on the graph and on the type of percolation process considered. It turns out that more can be said about the size of clusters in the subcritical phase. Let o ∈ V be an arbitrary, but fixed vertex in a quasi transitive graph G. Consider site or bond percolation on G and denote by Co (ω) the cluster containing o and by |Co (ω)| the number of vertices in Co (ω).

10


Theorem 4. For any p < pc there exist a constant τp > 0 such that P(|Co (ω)| ≥ n) ≤ e−τp n for all n ∈ N. In particular, the expected number of vertices in Co (ω) is finite for all p < pc . This extends earlier results of Aizenman & Newman [4], Menshikov [77] (see also [78]), and Aizenman & Barsky [2]. The generalisation to bond percolation on quasi transitive graphs is given in [8] and relies on the differential inequalities for order parameters established in [2]. The proofs of [8] apply to the site percolation process, too. In our later applications, we will need this result only for Cayley graphs of amenable finitely generated groups. 2.3. Percolation Hamiltonians on Cayley graphs We introduce percolation Hamiltonians with and without boundary terms and establish the existence of a selfaveraging IDS. This enables us to state our main result about the Lifshitz asymptotics and to compare it with the van Hove singularities of the Hamiltonian on the full Cayley graph. Consider an arbitrary subgraph G′ = (V ′ , E ′ ) of an infinite, k-regular Cayley graph G = (V, E). The multiplication operator W b.c. (G′ ) = k IdV ′ −D(G′ )

is non negative and has support in the interior vertex boundary of G′ . Thus ±W b.c. (G′ ) can be understood as a potential due to the repulsion/attraction of the boundary. When added to the adjacency Laplacian ∆A it gives rise to Dirichlet and Neumann boundary condition. Here we follow the nomenclature of [54, 81, 98]. Definition 5. We define the following operators on ℓ2 (V ′ ) (a) The Dirichlet Laplacian is defined as ∆D (G′ ) := ∆A + W b.c. (G′ ) = 2k IdV ′ −D(G′ ) − A(G′ ).

(b) The Neumann Laplacian is defined as

∆N (G′ ) := ∆A − W b.c. (G′ ) = D(G′ ) − A(G′ ). The Laplacian ∆(G) on the full graph will be abbreviated by ∆. Note that in this case all three versions of the Laplacian coincide since there is no boundary. Let us collect certain basic properties of the adjacency, Dirichlet and Neumann Laplacian. Remark 6. a) All operators introduced in Definitions 2 and 5 are bounded and selfadjoint. b) Since W b.c. (G′ ) ≥ 0 we have the following inequalities in the sense of quadratic forms: (6) ∆N (G′ ) ≤ ∆A (G′ ) ≤ ∆D (G′ ) This is complemented by 0 ≤ ∆N (G′ ) and ∆D (G′ ) ≤ 2k Id, which can be show by direct calculation in the same manner as in [54]. Later we will see, that there is a more complete chain of inequalities between five operators.


11

c) The operators ∆A (G′ ) and ∆D (G′ ) are injective. The Neumann Laplacian ∆N (G′ ) is injective if and only if G′ has no finite components. In particular, the dimension of the kernel of ∆N (G′ ) is the number of the finite components of G′ . Let us note that this relation between the number of zero Neumann eigenvalues and the number of clusters makes it possible to refine the analysis of the large time behaviour of random walks on finite percolation clusters, see, e.g., [100]. d) As we said in §1.4, it is a classical result [48] that zero is the lower spectral edge of ∆ if and only if G is an amenable graph. Finding lower bounds in the nonamenable case is a more difficult task, see for instance [10, 84, 117]. e) A graph G is called bipartite if there is a partition V1 , V2 of the vertex set such that there is no edge in G which joins two elements of Vi for i = 1, 2. For bipartite graphs it is useful to consider conjugation with the operator U on ℓ2 (V ) which is given by the multiplication with the function χV1 − χV2 . This operator is unitary, selfadjoint and an involution. We will denote the restriction of U to some subset ℓ2 (V ′ ) ⊂ ℓ2 (V ) again by the same symbol. The conjugation with U relates the upper spectral edge to the lower one, see, e.g., [54, 81, 98]. For any subgraph G′ we have D(G′ ) = UD(G′ ) U and A(G′ ) = −UA(G′ ) U. This implies the relations ∆A (G′ ) = 2k Id −U∆A (G′ ) U

N (D)

and ∆

(G′ ) = 2k Id −U∆D(N ) (G′ ) U.

(7)

Consequently the spectrum of ∆A (G′ ) is symmetric with respect to k, while the spectrum of ∆N (G′ ) is a set obtained from the spectrum of ∆D (G′ ) by the symmetry with respect to k. It follows that in the special case of amenable bipartite graphs the upper spectral edge of ∆ is equal to 2k. Bipartiteness is not only sufficient, but also necessary for the mentioned symmetry. Claim 4.5 from [89] shows that on amenable non-bipartite graphs 2k is not in the spectrum of ∆, thus the symmetry fails. Now we introduce percolation Laplacians associated to the configuration ω as bounded, selfadjoint operators on ℓ2 (V (ω)). Definition 7. Let (Ω, F, P) be a probability space corresponding either to site or bond percolation on G. For ω ∈ ΩV (resp. ω ∈ ΩE ) the operator # 2 2 ∆# ω := ∆ (G(ω)) : ℓ (V (ω)) → ℓ (V (ω)),

#

∈ {A, D, N }

is called adjacency, resp. Dirichlet, resp. Neumann percolation Laplacian. For brevity we introduce the notation Aω := A(G(ω))

and Wωb.c. := W b.c. (G(ω)).

Note that in the site percolation model we have dG(ω) (x) := ωx ωy , y∼x

12


while for bond percolation dG(ω) (x) := χV (ω) (x)

ωe

e∼x

where e ∼ x signifies that in the graph G the edge e is incident to x. Note that if the distance of x, y ∈ V is greater than two, the random variables dG(ω) (x) and dG(ω) (y) are independent. These facts imply that in both models the stochastic field (ω, x) → Wωb.c. (x) gives rise to a random potential which is stationary and ergodic with respect to the action of the group Γ. Due to the structure of the underlying percolation process the considered random operators satisfy an equivariance relation which we explain next. For a group Γ and an associated Cayley graph G there is a natural action of Γ on G by multiplication from the left. This gives rise to a Γ-action both on the corresponding percolation probability space and on the bounded operators on the ℓ2 space over the graph. The operation on the site percolation probability space (ΩV , PV ) by measure preserving transformations is given by (τγ (ω))x := ωγ −1 x , for γ ∈ Γ and x ∈ V . Note that γ and x are from the same set, but the first one is considered as a group element, while the second as a vertex. The family of transformations (τγ )γ∈Γ acts ergodically on (ΩV , PV ). In the same way there is an ergodic action of measure preserving transformations indexed by γ ∈ Γ, which we again denote by τγ , on the bond percolation probability space (ΩE , PE ). Here the transformations are given by (τγ (ω))e := ωγ −1 e , where γ −1 [y, z] = [γ −1 y, γ −1 z]. The group Γ acts by unitary translation operators (Uγ ϕ)(y) := ϕ(γ −1 y) on ℓ2 (V ). If we restrict these operators to the ℓ2 spaces over the active sites they act consistently with the shift on the probability space, more precisely U : ℓ2 (V (ω)) → ℓ2 (γV (ω)) = ℓ2 (V (τγ ω)). Due to the transformation behaviour of the adjacency operator Aω and the boundary term potential Wωb.c. one has the equivariance relation ∆# (τγ ω) = Uγ ∆# (ω)Uγ−1 for # ∈ {A, D, N }. Since (τγ )γ∈Γ acts ergodically on Ω, (∆# ω )ω falls into the class of random operators studied in [65]. This yields the non-randomness of the spectrum and spectral components. In the following let us denote by σ the spectrum, and by σdisc , σess , σpp , σsc , σc and σac , the discrete, the essential, the pure point, the singular continuous, the continuous, and the absolutely continuous component of the spectrum respectively. Let us recall that the pure point spectrum of an operator is the closure of the set of its eigenvalues. We use the notation σf in (H) for the set of eigenvalues of H which posses an eigenfunction with compact, i.e., finite, support. The following theorem holds for site and bond percolation Hamiltonians and for all values of p ∈ [0, 1]. Theorem 8. Let ∆# ω be one of the percolation Laplacians defined above. Then there ′ exists for each • ∈ {disc, ess, pp, sc, c, ac, fin} a subset Σ# • ⊂ R and a subset Ω ⊂ # ΩV (resp. Ω′ ⊂ ΩE ) of full measure, such that σ• (∆# ω ) = Σ• holds for every # ω ∈ Ω′ and for all • ∈ {disc, ess, pp, sc, c, ac, fin}. In particular σ(∆# ω ) = Σ for all ω ∈ Ω′ .


13

This has been proven for site percolation models in [65, 111, 112], and holds with the same proofs for bond percolation. For similar results in the case of bond percolation on the lattice see also [54]. Remark 9. (a) All the results stated so far hold for arbitrary quasi transitive graphs. In particular it is not necessary to assume that the graph is amenable or infinite. For infinite graphs the discrete spectrum Σdisc is empty. 2 (b) The percolation Laplacians ∆# ω are defined on ℓ (V (ω)). Thus, technically (∆# ) is associated to a direct integral operator with non-constant fibre. One ω∈Ω ω can modify the operator in such a way that the fibres become constant and that the spectrum changes in a controlled way. First note that for a fixed configuration ω and a fixed cluster C(ω) any percolation Hamiltonian leaves ℓ2 (C(ω)) invariant. Consequently, the full operator decomposes according to the clusters: ∆# ∆# (C(ω))) (8) ω = C(ω) cluster of G(ω)

For any constant K we may add to (8) the operator K Idx = K IdV \V (ω) . x∈V \V (ω)

2 # Denote the resulting sum by ∆ ω . It acts on ℓ (V ) and leaves the subspaces # ℓ2 (V (ω)) and ℓ2 (V \ V (ω)) invariant. In particular, ∆ ω can be written in the ˜ same way as (8) with the only difference that the direct sum extends over all C(ω) which are either a cluster in G(ω) or a vertex in V \ V (ω). If the configuration ω # # is such that V (ω) = V then ∆ ω coincides with ∆ω . Note however that these configurations form a set of measure zero, both in the site and the bond percolation model. In all other cases we have # # σ(∆ ω ) = σ(∆ ) ∪ {K}, ω

# # σpp (∆ ω ) = σpp (∆ω ) ∪ {K} # # σc (∆ ω ) = σc (∆ω ).

and

(c) In the subcritical regime there are no infinite clusters. The decomposition (8) implies that the operator is a direct sum of finite dimensional operators, thus the spectrum consists entirely of eigenvalues in σfin (∆# ω ). In particular, Σsc = Σac = ∅ and Σpp = Σfin . (d) We are not able to calculate the deterministic spectrum Σ# but only to give partial information about it. Since we can find arbitrarily large clusters in G(ω) almost surely, strong resolvent convergence gives that Σ# ⊃ σ(∆), see [54] for details. Together with σ(∆# ω ) ⊂ [0, 2k] for all ω this implies that in the case of amenable Cayley graphs zero is the lower spectral edge of the operators ∆# ω almost surely for any # ∈ {A, D, N }. Similar inclusions for the deterministic spectrum of Anderson models have been given in [63]. If the Cayley graph is bipartite and the spectrum of the free Laplacian ∆ has no gaps we have Σ# = [0, 2k]. If σ(∆)

14


has several components, it might be that Σ# contains values in a spectral gap of ∆. This is related to the phenomenon called spectral pollution encountered when using strong convergence to approximate spectral values of the limiting operator, see for instance [25, 68] and the references therein. Our next aim is to introduce the IDS and state its main properties. An abstract IDS may be defined without an exhaustion procedure by setting N # (E) = E{χ]−∞,E] (∆# ω )δι , δι } see [65]. However, the physically interesting situation is when the IDS is selfaveraging, i.e., when spatial averages of the normalised eigenvalue counting functions converge to the expectation with respect to the randomness. To be able to show that the IDS has indeed this property we have to require that the group Γ is amenable. Each such group has a sequence of finite, non-empty subsets (Ij )j which satisfies for every finite set F ⊂ Γ the property lim

j→∞

|Ij △ F · Ij | = 0. |Ij |

Such an (Ij )j is called a Følner sequence. Here △ denotes the symmetric difference of two sets. Various properties of such sequences are discussed in [1], and their role in the construction of the IDS of random operators in [95] and [110, §2.3]. Each Følner sequence has a tempered subsequence (Ijk )k , meaning that there exists a | ≤ C|Ijk |. We may consider Ijk as a subset of the verconstant C such that |Ijk Ij−1 k−1 tices in the Cayley graph. Denote by Pk the projection on the vertex set Ijk ∩V (ω) # ∗ and by ∆#,k ω the restricted operator Pk ∆ω Pk for any # ∈ {A, D, N }. Thus we obtain a selfadjoint operator on a finite dimensional space which has a finite number of real eigenvalues. Hence the trace of the spectral projection Tr χ]−∞,E] (∆#,k ω ) is finite. We define the eigenvalue counting distribution function as 1 Tr χ]−∞,E] (∆#,k Nω#,k (E) := ω ) . |Ik |

The next theorem holds for site and bond percolation Hamiltonians and for all values of p ∈ [0, 1].

Theorem 10. Let G be a Cayley graph of an amenable group Γ and (Ik )k a tempered Følner sequence. There exists a set Ω′ ⊂ ΩV (resp. ΩE ) of full measure such that for every ω ∈ Ω′ and every E ∈ R which is a continuity point of N # (·) we have lim Nω#,k (E) = N # (E).

k→∞

The support of the measure associated to the distribution function N # equals Σ# . For site percolation, the theorem is proven in [111], the modification to bond percolation is not hard. See also [54] for the case of bond-percolation on the lattice. Thus we have obtained an selfaveraging IDS. The theorem implies that zero is the lower edge of the support of N # for # ∈ {A, D, N } and that N A (0) =


15

N D (0) = 0 while N N (0) > 0. The distribution function N # has a discontinuity at E ∈ R if and only if E ∈ Σ# fin . This will be discussed in more detail in §3.1.

Let us denote the density of vertices in V \ V (ω) by g(ω). By ergodicity there is a value g ∈ [0, 1] such that g(ω) = g almost surely. In the site percolation model # g = 1 − p and in the bond percolation model g = (1 − p)k . Then the IDS N of # # (∆ω )ω is related to N by # = N # + g χ[K,∞[ . N

Remark 11. Let µ be a non-trivial probability measure on a compact interval [0, s] and (wω (x))x∈V an i.i.d. family of random variables with distribution µ. The Anderson model on the graph G is given by Hω := ∆ + Wω : ℓ2 (V ) → ℓ2 (V ). Here Wω is the multiplication operator with wω (x). This is again an ergodic operator with non-random spectrum and selfaveraging IDS. It is very convenient to compare it with the site percolation Laplacian. For this purpose we specialise to the case µ = pδ0 + (1 − p)δ1 and introduce an additional, global coupling constant λ ≥ 0. BA BA We denote the quadratic form of the operator Hω,λ := ∆ + λWω by qω,λ . Here BA stands for the Bernoulli-Anderson or binary-alloy type of the Hamiltonian. Denote its IDS by NλBA . Recall that one may compare two (lower-bounded, closed) quadratic forms q1 and q2 on a Hilbert space by setting q1 ≤ q2 if and only if the domains satisfy the inclusion D(q1 ) ⊃ D(q2 ) and for all ϕ ∈ D(q2 ) we have q1 [ϕ] ≤ q2 [ϕ]. This notion of inequality of quadratic forms is consistent with extending each form to the whole Hilbert space by the rule qi [ϕ] = +∞ for ϕ ∈ D(qi ). Denote the quadratic form associated to any of the Hamiltonians ∆# ω by qω# . Comparing the Anderson and the adjacency site percolation Hamiltonian we BA ≤ qωA for all λ ≥ 0. This relation holds pointwise for all ω if we see that qω,λ introduce the obvious coupling between the random potential (wω (x))x∈V and BA converges the percolation process. Moreover for λ → ∞ the quadratic form qω,λ monotonously to qωA . On the other hand, since the potential (wω (x))x∈V is nonBA where qG denotes the quadratic form of ∆ on the full negative we have qG ≤ qω,λ N N graph. Using the argument in [98, §2] one sees that q ω ≤ qG , where qω is the form N N corresponding to the operator ∆ = ∆ ⊕ 0 · Id . V \V (ω) ω ω Summarising all quadratic form inequalities obtained so far for the site percolation case we obtain the chain BA A D N q ω ≤ qG ≤ qω,λ ≤ qω ≤ qω

which implies for the corresponding IDS’

N N (E) ≥ NG (E) ≥ NλBA (E) ≥ N A (E) ≥ N D (E) for all E ∈ R.

Now it is clear why the study of the Anderson Hamiltonian in [16] gave also results on the (adjacency) percolation Laplacian. Upper bounds on the IDS NλBA imply upper estimates for the IDS of the adjacency and Dirichlet site percolation model.

16


There is also a relation between bond percolation Hamiltonians on the one hand and random hopping models [58] and discrete Schr¨ odinger operators with random magnetic field [85] on the other hand. All three models share the feature that the randomness enters the Hamiltonian in the off-diagonal matrix elements. In fact, in [58, 85] certain spectral properties of such models are analysed by relating them to Hamiltonians with diagonal disorder. Remark 12. The Anderson model on ℓ2 (Zd ) is interpreted by physicists as a single spectral band approximation of a continuum random Schr¨ odinger operator on L2 (Rd ). Due to the i.i.d. assumption on the random variables in the Anderson model its spectrum as a set equals almost surely σ(∆) + supp µ := {t + s ∈ R | t ∈ σ(∆), s ∈ supp µ}

see Theorem III.2 in [63]. Here as before µ denotes the distribution measure of the potential values of the Anderson model and supp µ its (topological) support. Now, if the support of µ is split into several connected components and if the gaps between them are large enough, the a.s. spectrum of the Anderson model also contains gaps and consequently has a (sub)band structure. Internal Lifshitz tails for such models have been studied in [81, 99]. BA introduced In particular, if we consider the Bernoulli-Anderson model Hω,λ BA splits into above, we see that for large enough λ the almost sure spectrum of Hω,λ subbands. As we let λ → ∞, one of the bands diverges to infinity. On the other BA hand, we know that in the sense of quadratic forms qω,λ ր qωA for λ → ∞. Thus the resulting site percolation Hamiltonian ∆A ω may be understood as an approximation BA (with large values of λ) associated to a spectral (sub)band. of Hω,λ Now we are in the position to state our main results in the next two theorems. In both cases we have the same setting as before: we consider a Cayley graph G of an amenable finitely generated group Γ, site or bond percolation on G and the # percolation Laplacians ∆# ω with associated IDS N . We restrict ourselves now to the subcritical phase p < pc . The asymptotic behaviour of the IDS of the adjacency and the Dirichlet percolation Laplacian at low energies is as follows: Theorem 13. Assume that G has polynomial growth and V (n) ∼ nd . Then there − are positive constants α+ D (p) and αD (p) such that for all positive E small enough −

e−αD (p)E

−d/2

+

≤ N D (E) ≤ N A (E) ≤ e−αD (p)E

−d/2

Assume that G has superpolynomial growth. Then ln | ln N A (E)| ln | ln N D (E)| lim = lim = ∞. Eց0 Eց0 | ln E| | ln E|

.

(9)

(10)

In particular, the IDS is very sparse near E = 0 and consequently the bottom of the spectrum is a fluctuation boundary. Relations (4) and (9) imply that in the case of polynomial growth the Lifshitz exponent coincides with the van Hove exponent of the Laplacian on the full Cayley graph. In the case of superpolynomial growth we have that both exponents are infinite. One may ask whether the limits


17

(5) and (10), which define the two exponents, diverge at the same rate. To express this in a quantitative way let us note that in the case of polynomial growth we have ln | ln N A (E)| =1 lim Eց0 | ln N0 (E)|

and the analogous relation for the Dirichlet Laplacian. In the case of superpolynomial growth one may hope to prove at least lim

Eց0

ln ln | ln N A (E)| = 1. ln | ln N0 (E)|

We are not able prove that in general, but at least for the case of the Lamplighter group, see §3.3. N In the case of Neumann boundary conditions we know that ∆ ω ≤ ∆. ConN is at the bottom of the spectrum at least as ‘thick’ as the sequently, the IDS N one of the free Laplacian, which exhibits a van Hove singularity. In particular, the N energy zero is not a fluctuation boundary. This remains true if we pass from N to N N by removing the point mass gδ0 from the density of states measure. − Theorem 14. There exist positive constants α+ N (p) and αN (p) such that for all positive E small enough −

e−αN (p)E

−1/2

+

≤ N N (E) − N N (0) ≤ e−αN (p)E

−1/2

.

(11)

The value N (0) coincides with the average number of clusters per vertex in the random graph G(ω). After subtracting this value we can speak of (11) as a kind of ‘renormalised’ Lifshitz asymptotics with exponent 1/2. N

Remark 15. (a) In the next section we give a sketch of the proof of the theorems, while the full version will appear elsewhere. (b) The lower bounds in (9) and (11) are actually true for all values of p. (c) In the special case of bipartite Cayley graphs there is a relation between the behaviour of the IDS near the upper and lower spectral edges of the spectrum, see Remark 6. In that case we are able to characterise the asymptotic behaviour of the IDS at the upper spectral boundary in the same way as done in [54].

3. Discussion, additional results and sketch of proofs In this final section we conclude the paper with a discussion of further spectral properties of percolation Hamiltonians, an outline of the proof of the Theorems 13 and 14, and an important example. The example concerns the Lamplighter group, which is amenable, but of exponential growth. The spectral properties we mentioned are related to jumps of the IDS, to finitely supported eigenfunctions and to the unique continuation principle, see [112, 114] and [66].

18


3.1. Discontinuities of the IDS Let us recall a result on the location of the set of discontinuities of the IDS, established in [112] for site percolation Laplacians and generalisations thereof. The result and its proof apply verbatim for the bond percolation model. Furthermore for this result it is not necessary to assume that the percolation process is independent at different sites, but merely that it is ergodic with respect to the transformations (τγ )γ∈Γ introduced in §2.2. Proposition 16. Let G be a Cayley graph of an amenable group Γ and (∆# ω )ω , # ∈ {A, D, N } a percolation Laplacian associated either to site or to bond percolation. Then the following two properties are equivalent: (i) the IDS of (∆# ω )ω is discontinuous at E ∈ R, (ii) E ∈ Σ# fin . Remark 17. This result can be strengthened to describe the size of the jumps also. In a forthcoming paper [66] of Lenz and the second named author ergodic Hamiltonians on discrete structures are analysed. The abstract setting considered there covers in particular Anderson Hamiltonians and site and bond percolation Laplacians (∆# ω )ω on a Cayley graph G of an amenable group Γ. Again one has to assume that the stochastic process which determines the random potential, respectively the percolation process entering the percolation Laplacian, is ergodic with respect to the transformation group (τγ )γ∈Γ . Then the eigenspace corresponding to the eigenvalue E is spanned by finitely supported eigenfunctions almost surely. See also [61, 62] for a similar result for periodic operators on graphs with an Abelian group structure. The jumps of the IDS of invariant operators on Cayley graphs have been studied in the literature on L2 -invariants. The kth L2 -Betti number is the trace per unit volume of the kernel of the Laplacian on k-forms. It has been proven in great generality that it does not matter whether one defines it via the continuum or the combinatorial Laplacian, see, e.g., [30]. Likewise it is known that the zeroth L2 -Betti number vanishes on amenable Cayley graphs [74, Thm. 1.7], which is the same as saying that the IDS is continuous at the bottom of the spectrum. Using the same terminology for random ergodic operators, we may say that the Neumann percolation Laplacian has non vanishing (zero order) L2 -Betti number. With this regard it would be interesting to find an interpretation of higher order Betti numbers in terms of quantities of mathematical physics. The sizes of jumps of the IDS at discontinuity points have be discussed for Abelian periodic models in [34, 104]. The task of characterising the sizes of jumps of the IDS is related to the Atiyah conjecture, see, e.g., [39, 75] or [74, Conjecture 2.1], and the references therein. 3.2. Outline of the proof of Theorems 13 and 14 While the full proofs of our results will be given in [9] we present here certain key estimates. In particular, we state upper and lower bounds for the IDS of percolation Laplacians in a neighbourhood of the lower spectral edge. The bounds are given


19

in terms of λ# (G′ ), the lowest non-zero eigenvalue of ∆# (G′ ). These estimates are generalisations of Lemmata 2.7 and 2.9 in [54]. The proof of Theorem 18 uses the exponential decay from Theorem 4. Theorem 18. Let G be an amenable Cayley graph and # ∈ {A, D, N }. Assume that there is a continuous strictly decreasing function f : [1, ∞[ → R+ such that lims→∞ f (s) = 0 and λ# (G′ ) ≥ f (|G′ |) for any finite subgraph G′ . Then, for every 0 < p < pc the IDS satisfies the following inequality N # (E) − N # (0) ≤ e−ap f

−1

(E)

,

(12)

for some positive constant ap , when E is small enough. Here f inverse function of f .

−1

denotes the

Theorem 19. Let G be an amenable Cayley graph and # ∈ {A, D, N }. Suppose that there is a sequence of connected subgraphs (G′n )n and a sequence (cn )n in R+ such that (i) limn→∞ |G′n | = ∞ (ii) limn→∞ cn = 0 (iii) λ# (G′n ) ≤ cn

For every E > 0 small enough define n(E) := min {n; cn ≤ E}. Then for every 0 0 small enough ′

N # (E) − N # (0) ≥ e−bp |Gn(E) | .

(13) ′

Hence our problem is reduced to finding efficient bounds for λ (G ) in terms of the geometric properties of G′ . For this we will use, following [54], the Cheeger and Faber & Krahn inequalities. Since we are considering general Cayley graphs of amenable groups with polynomial growth, these two inequalities are not sufficient, but we will need additionally an appropriate version of the isoperimetric inequality. The function V was defined in Theorem 1. We also define #

φ(t) := min {n ≥ 0; V (n) > t} . Moreover we will denote the linear subgraph with n vertices by Ln . Proposition 20. For a Cayley graph G = (V, E) there are positive constants αD , βD , γD , αN and γN such that the following are true (i) For every finite connected subgraph G′ αD and λA (G′ ) ≥ φ(βD |G′ |)2

λN (G′ ) ≥

αN . |G′ |2

(14)

(ii) For every positive integer n λD (B(n)) ≤

γD V (n) n2 V (⌊n/2⌋)

and

λN (Ln ) ≤

γN . n2

(15)

20


Sketch of the proof. (i) The inequality for adjacency Laplacian can be proved following the arguments in the proofs of Lemma 2.4 from [54] and Proposition 7.1 from [19] and using the isoperimetric inequality from Théorème 1 in [22]. The inequality for Neumann Laplacian is a simple consequence of the Cheeger inequality (see Théorème 3.1 in [21]). (ii) Both inequalities can be obtained by inserting an appropriate test function to the sesquilinear form and using the mini-max principle. For the Dirichlet Laplacian we choose a test function which has value 0 outside the ball B(n), value i on the sphere of radius n−i, for i = 0, . . . , ⌈n/2⌉ and value ⌈n/2⌉ inside the ball B(⌊n/2⌋). The operator ∆N (Ln ) is not injective and thus the test function must be orthogonal to the kernel. This means that the test function ϕ must have support inside Ln and x∈Ln ϕ(x) = 0. The appropriate test function is the one which grows to n−1 linearly from −n+1 2 2 along the vertices of Ln and is 0 outside of Ln . Remark 21. (a) In the case of polynomial growth the first bound in (14) and the first bound in (15) are of the same order in n if G′ = B(n). Likewise the second bounds in (14) and (15) are of the same order of magnitude for G′ = Ln . Thus the proposition shows that the optimal subgraph configuration for adjacency and Dirichlet Laplacians is a ball, and for the Neumann Laplacian is a line graph. For Dirichlet and Neumann boundary conditions this can be motivated in terms of the potential ±W b.c. . Since W b.c. is non-negative (repulsive) at the interior vertex boundary, a test function ϕ with low expectation value ϕ, ∆D (G′ )ϕ has to make both ϕ, ∆A (G′ )ϕ and ϕ, W b.c. ϕ small. The second condition pushes the mass of ϕ away form the boundary, while the first one minimizes the variation of ϕ. This is best realised when G′ is a ball. On the contrary, in the case of Neumann boundary conditions −W b.c. ≤ 0 is attracting ϕ to the boundary. For fixed volume the graph with most boundary is a linear graph. (b) Note that if Γ is a group of polynomial growth, then the balls B(n), n ∈ N form a Følner sequence: Since every finite F satisfies F ⊂ B(k) for some k ∈ N we have |F · B(n) \ B(n)| ≤ V (n + k) − V (n) ≤ (c + o(1)) (n + k)d − (c − o(1)) nd = o(V (n)) and similarly |B(n) \ F · B(n)| ≤ V (n) − V (n − k) = o(V (n)). Since V (2n) ≤ 2d ab V (n) the sequence of balls satisfies the doubling property and is in particular tempered. Note that although every amenable group contains a Følner sequence, balls may not form one. 3.3. The Lamplighter group In this section we will explain how the ideas and methods we used to study the behaviour of the IDS in the case of groups of polynomial growth, can be used to give sharp bounds on the IDS asymptotics in the case of a particular group of superpolynomial growth. Namely, we consider certain Lamplighter groups, which are examples of amenable groups with exponential growth (i.e., there exists a constant c > 1 such that V (n) ≥ cn ).


21

We define the Lamplighter group as the wreath product Zm ≀Z, where m is an arbitrary positive integer. Elements of this group are ordered pairs (ϕ, x), where ϕ is a function ϕ : Z → Zm with finite support and x ∈ Z. The multiplication is given by (ϕ1 , x1 ) ∗ (ϕ2 , x2 ) := (ϕ1 + ϕ2 (· − x1 ), x1 + x2 ). More generally, a lamplighter group can be defined as a wreath product of two Abelian groups, one of which is finite. However, here we shall only consider lamplighter groups of the form Zm ≀ Z. Since Lamplighter groups are amenable it makes sense to consider the integrated density of states in both the deterministic and random setting, as defined in Section 2. The next theorem concerns the Laplace operator on the full Cayley graph. It establishes a Lifshitz-type asymptotics in the sense that ln | ln N0 (E)| 1 = . | ln E| 2

lim

Eց0

(16)

Of course the underlying operator is not random, but the IDS is exponentially thin at the bottom of the spectrum as it is the case for spectral fluctuation boundaries of random operators. The quantity (16) may be called a Lifshitz exponent or maybe more appropriately secondary Novikov-Shubin invariant of the Laplacian, using the terminology of [89]. Theorem 22. Let G be a Cayley graph of the Lamplighter group. There are positive + constants a+ 1 and a2 such that +

−a2 E N0 (E) ≤ a+ 1e

−1/2

, for all E small enough.

− Moreover for every r > 1/2 there are positive constants a− r,1 and ar,2 such that −

−ar,2 E N0 (E) ≥ a− r,1 e

−r


The proof of the preceding theorem follows from the proof of Theorem 4.4 (parts (ii) and (iii)) in [89]. As an input we need the following inequalities for µ(2n) (ι), the return probability of the simple random walk after 2n steps: 1/3

a1 e−a2 n

1/3

≤ µ(2n) (ι) ≤ A1 e−A2 n

,

for some positive constants ai , Ai , i = 1, 2, and all positive integers n. For the reference see Theorem 15.15 in [115]. For the percolation case we shall once again use Theorems 18 and 19. Exponential growth and Proposition 20 i) give lower bounds for the lowest eigenvalues λA (G′ ) which are of the form const /(ln |G′ |)2 . Now Theorem 18 implies the following result. Theorem 23. Let G be an arbitrary Cayley graph of the lamplighter group Zm ≀ Z. For every p < pc there are positive constants b1 and b2 such that the IDS of the adjacency and Dirichlet percolation Laplacian satisfies the following inequality b2 E −1/2

N A (E) ≤ N D (E) ≤ e−b1 e


(17)

22


The upper bounds from Proposition 20 are not applicable in the Theorem 19 any more. Instead, we shall use results from [11], where Bartholdi & Woess proved that the Laplacian ∆ on Diestel-Leader graphs has only pure point spectrum, ˙ expanding on earlier results of Grigorchuk & Zuk [40] and Dicks and Schick [29] concerning certain Cayley graphs of Lamplighter groups. Diestel-Leader graphs are a generalisation of Cayley graphs of Lamplighter groups with a particular set of generators. This ‘natural’ set of generators is given by S0 := {(l · δ1 , 1), l ∈ Zm } ∪ {(l · δ0 , −1), l ∈ Zm } .

(18)

Here l · δz denotes the function which has value l in z and 0 everywhere else. In the following we formulate certain facts about the spectrum of adjacency Laplacians on certain finite, connected subgraphs of the Lamplighter graph, called tetrahedrons, which were established in [11], see also [29, 40]. We will not give a definition of these subgraphs, since it would require a quite comprehensive description of horocyclic products of homogeneous trees. Rather, we refer to [11] for precise definitions and background information. We will need three facts concerning tetrahedrons and corresponding eigenvalues of adjacency Laplacians: (a) the tetrahedron of depth n (denoted by Tn ) has (n + 1)mn vertices, (b) 2m(1 − cos nπ ) is an eigenvalue of the operator ∆A (Tn ), (c) there is an eigenvector of ∆A (Tn ), corresponding to the eigenvalue 2m(1 − cos πn ), that has value 0 on the inner vertex boundary of Tn . Claim a) follows directly from the definition of tetrahedron. For the proofs of claims b) and c) see Lemma 2 and Corollary 1 in [11]. Now Theorem 19, with G′n = Tn and cn = 2m(1−cos nπ ), implies the following bound. Theorem 24. Let GS0 be the Cayley graph of the lamplighter group Zm ≀ Z defined with respect to the set of generators S0 . For every 0 < p < 1 there are positive constants c1 and c2 such that the IDS of percolation Laplacians satisfies the following inequality c2 E −1/2

e−c1 e

≤ N D (E) ≤ N A (E), for all E small enough.

(19)

Theorem 24 is in fact valid for all Cayley graphs of the lamplighter group Zm ≀ Z (with constants c1 and c2 possibly depending on the choice of the generator set). This can be seen as follows: Assume we are given a Cayley graph GS defined with respect to a generator set S. A natural candidate for the sequence of subgraphs G′n in Theorem 19 would be GS (Vn ), the subgraphs in GS induced by Vn (Vn being the vertex set of a tetrahedron with depth n). Since this subgraph is not necessarily connected, to each vertex x in Vn we add a ball (in GS ) of some large, but fixed radius R, centered at x. In this way we get the vertex set VnR , which is connected in GS . Because of fact (c) above, the adjacency Laplacian on the subgraph of GS0 induced by the vertex set VnR is again bounded above by 2m(1 − cos nπ ). Now calculations similar


23

to those in the proof of Theorem 3.2 in [115] show that the lowest eigenvalue of the adjacency Laplacian on the GS (VnR ) has an upper bound of the form const(1− cos πn ). Thus Theorem 19 with G′n = GS (VnR ) proves the claim in Theorem 24 for the Cayley graph GS . Acknowledgements It is a pleasure to thank D. Lenz, I. Nakić and F. Sobieczky for stimulating discussions related to the topics of this paper. Visits of the authors to the TU Graz ¨ were supported by the ESF through a Short Visit Grant and the Ostereichischer Fonds zur F¨ orderung der wissenschaftlichen Forschung.

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On Inverse Problems for Finite Trees B.M. Brown and R. Weikard Abstract. In this paper two classical theorems by Levinson and Marchenko for the inverse problem of the Schr¨ odinger equation on a compact interval are extended to finite trees. Specifically, (1) the Dirichlet eigenvalues and the Neumann data of the eigenfunctions determine the potential uniquely (a Levinson-type result) and (2) the Dirichlet eigenvalues and a set of generalized norming constants determine the potential uniquely (a Marchenko-type result). Mathematics Subject Classification (2000). Primary 34A55, Secondary: 05C05. Keywords. Dirichlet-to-Neumann map; inverse spectral problems; trees.

1. Introduction Some 60 years ago Borg, Levinson, and Marchenko established the now famous inverse spectral theory for the Schr¨ odinger equation on a compact interval. Denote the solution of the initial value problem −y ′′ + qy = λy, y(0) = 0 and y ′ (0) = 1 on the interval [0, 1] by s(λ, ·). Then the real integrable potential q is uniquely determined by any of the following sets of data: • Borg [5] (1946): The Dirichlet-Dirichlet eigenvalues and the Dirichlet-Neumann eigenvalues, i.e., the zeros of s(·, 1) and of s′ (·, 1). • Levinson [17] (1949): The Dirichlet-Dirichlet eigenvalues (denoted by λn ) and the Neumann data of the associated eigenfunctions, i.e., the numbers s′ (λn , 1) (recall that s′ (λn , 0) = 1). • Marchenko [18] (1950): The Dirichlet-Dirichlet eigenvalues and the norming

1 constants 0 s(λn , t)2 dt of the associated eigenfunctions. Since the 1980s spectral problems on graphs and trees have also been investigated (we are interested here solely in so-called metric trees where edges are homeomorphic to real intervals and may support potentials). We refer the reader This material is based upon work supported by the National Science Foundation under Grant No. DMS-0304280.

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B.M. Brown and R. Weikard

to the excellent surveys [13] and [14] by Kuchment for an overview of these developments and their applications. Investigation of inverse problems on such graphs and trees are not quite so numerous. Without a claim to completeness we list here the works of Gerasimenko [11]; Curtis and Morrow [9], [10]; Carlson [7]; Pivovarchik [21], [20]; Kurasov and Stenberg [15]; Belishev [2]; Brown and Weikard [6]; Harmer [12]; Kurasov and Nowaczyk [16]; Yurko [22]; Belishev and Vakulenko [3]; and Avdonin and Kurasov [1]. We emphasize that the results in [2], [22], and [1] are particularly close to ours. In this paper we address generalizations of the above mentioned results of Levinson and Marchenko to the case of finite trees with r edges and n0 boundary vertices. The generalization of Borg’s result has been established recently by Yurko [22]: spectra for n0 (specific) boundary conditions are sufficient to determine the potential on the tree. In this paper we focus for simplicity on real and bounded potentials on trees whose edge lengths are one. Our methods generalize to the case of integrable complex potentials on trees with varying edge lengths. We plan to address such generalizations in a subsequent paper. Under the given circumstances the Dirichlet eigenvalues, which we denote by λ1 , λ2 , . . . , are real and discrete since they are zeros of an entire function which can not vanish away from the real axis. We use the letter Σ to signify the set of all Dirichlet eigenvalues. Geometric multiplicities of Dirichlet eigenvalues may be larger than one. The multiplicity of the eigenvalue E is denoted µ(E). The generalization of Levinson’s result reads as follows. Theorem 1.1. Let q be a real-valued bounded potential supported on a finite metric tree whose edge lengths are one. Then the Dirichlet eigenvalues, their multiplicities, and the Neumann data of an orthonormal set of Dirichlet eigenfunctions uniquely determine the potential almost everywhere on the tree. We remark here that eigenfunctions are not automatically orthonormal since eigenspaces may have dimensions larger than one. Theorem 1.1 will be proved in Section 6. A particularly important role is played in this paper by the Weyl solutions. They are uniquely defined away from the Dirichlet eigenvalues by requiring that they satisfy homogeneous Dirichlet boundary conditions at all but one boundary vertex where they assume the value 1. At the Dirichlet eigenvalues the Weyl solutions ∞ cease to exist but multiplying them with the “minimal function” χ(λ) = E∈Σ (1 − λ/E) (as opposed to the characteristic function which takes the multiplicities of the eigenvalues into account) gives globally defined functions ω(k, λ, ·) where k ∈ {1, . . . , n0 } signifies the boundary vertex at which the function value is prescribed to be χ(λ). If E is a Dirichlet eigenvalue of multiplicity µ(E) then there will be µ(E) of the functions ω(k, λ, ·) which are linearly independent. Let K(E) ⊂ {1, . . . , n0 } be a maximal set such that {ω(k, λ, ·) : k ∈ K(E)} is linearly independent. We also introduce the quantities Nj,k (E) which are the inner

On Inverse Problems for Finite Trees

33

products of the functions ω(j, E, ·) and ω(k, E, ·). For j = k these are norming constants of eigenfunctions. We can now formulate the generalization of Marchenko’s result. Theorem 1.2. Let q be a real-valued bounded potential supported on a finite metric tree whose edge lengths are one. Then the Dirichlet eigenvalues, their multiplicities, and the quantities Nj,k (E), j ∈ {1, . . . , n0 }, k ∈ K(E), E ∈ Σ uniquely determine the potential almost everywhere on the tree. This theorem will be proved in Section 7. The paper is organized as follows. Section 2 gives formal definitions of trees, interface conditions, and operators under consideration. It also provides basic results on initial value and Dirichlet boundary value problems. Weyl solutions and Green’s function play (as is to be expected) an important role. They are studied in Section 3 and Section 4, respectively. The Dirichlet-to-Neumann map is introduced in Section 5. Sections 6 and 7 are devoted to the proofs of Theorems 1.1 and 1.2, respectively.

2. Preliminaries 2.1. Trees A finite tree is given by a Hausdorff space T and a set of r homeomorphisms ǫj : [0, 1] → T , j = 1, . . . , r, such that the following conditions are met: 1. T = rj=1 {ǫj (t) : t ∈ [0, 1]}. 2. If ǫj (t) = ǫk (s) and j = k then t, s ∈ {0, 1}. 3. T is simply connected. In this paper we will be concerned only with finite trees and we will always mean a finite tree when we speak of a tree. The set V = {ǫ1 (0), ǫ1 (1), . . . , ǫr (0), ǫr (1)} has precisely r + 1 elements called vertices of the tree. The homeomorphisms ǫj are called edges of the tree. We say a vertex v belongs to an edge ǫj or that edge ǫj is attached to v if v = ǫj (0), the initial vertex of ǫj , or v = ǫj (1), the terminal vertex of ǫj . A vertex is called a boundary vertex if it belongs to only one edge. Such an edge will be called a boundary edge. A vertex which belongs to several edges is called an internal vertex. An edge both of whose endpoints are internal vertices is called an internal edge. The fact that edges are homeomorphic to the metric space [0, 1] turns the space T in a natural way into a metric space. Since the interval [0, 1] has an orientation associated with it, so does each of the edges. However, we are not interested in the orientations of the edges, but only in the metric structure provided by the homeomorphisms ǫj . Thus, given a finite tree we obtain another tree by replacing the homeomorphism t → ǫj (t) by t → ǫj (1 − t) for some (or none or all) of the indices j. The new tree is, of course, still associated with the Hausdorff space T and the trees, so related, are called equivalent. This is an equivalence relation and we will henceforth choose

34


the labels and orientations for the edges as convenience suggests. Often times we will call the Hausdorff space T a tree, assuming a tacit understanding of its metric structure. We take here (for the most part) the point of view that all boundary vertices play identical roles. Assuming there are n0 boundary vertices we will then assign labels 1, . . . , n0 to the boundary vertices and the corresponding boundary edges. We assume also that the boundary edges are oriented so that vj = ǫj (0) for j = 1, . . . , n0 . However, in some intermediate results one boundary vertex of the tree is singled out. In this case it is convenient to assign labels and orientations in the following way. The special boundary vertex is called the root and is designated as v0 . All other boundary vertices are called branch tips. The edge attached to the root, denoted by ǫ0 , will be called the trunk. All edges are oriented so that their initial vertex is closer to the root than their terminal vertex. In particular, ǫ0 (0) = v0 , the root of the tree. The edges other than the trunk attached to the terminal vertex of the trunk v1 = ǫ0 (1) are called limbs. Each one of them is the trunk of a subtree with root v1 . A tree labeled and oriented this way will be called a rooted tree. We emphasize that all results obtained are independent of the particular way in which edges and vertices are labeled and in which edges are oriented; these designations only serve to communicate proofs more easily. 2.2. The interface conditions A function y defined on T may be represented as y = (y1 , . . . , yr )⊤ where yj (t) = y(ǫj (t)). We say that y is integrable on T (or square integrable on T or y ∈ Lp (T )) if the yj are integrable on [0, 1] (or square integrable on [0, 1] or yj ∈ Lp ([0, 1])) for j = 1, . . . , r. We define C to be the set of all functions y defined on T which satisfy the following conditions: 1. For each j the functions yj and yj′ are absolutely continuous on [0, 1]. 2. y is continuous on T . 3. For each internal vertex v the Kirchhoff condition yj′ (0) = 0 yj′ (1) − ǫj (1)=v

ǫj (0)=v

holds. Conditions 2 and 3 are called interface conditions. Note that C is independent of the orientation or the labeling of the edges. In order to deal conveniently with the interface conditions we introduce1 the operator I = E0 E0 + E1 E1 + D0 D0 + D1 D1 , 1 This

notation is different from the one we used in [6]


35

where E0 , E1 , D0 and D1 are certain (2r − n0 ) × r matrices whose entries are 0 or ±1 and E0 , etc. are evaluation operators defined by E0 y = y(0), E1 y = y (1), D0 y = y ′ (0), and D1 y = y ′ (1). Thus y satisfies the interface conditions if and only if Iy = 0. 2.3. The differential equations If q is an integrable function on T define Q = diag(q1 , . . . , qr ). We will consider the differential expression y → −y ′′ +Qy and the differential equation −y ′′ +Qy = λy . Define D = {y ∈ C : −yj′′ + qj yj ∈ L2 ([0, 1])}.

We now define the operator L : D → L2 (T ) by (Ly)(ǫj (t)) = −yj′′ (t) + qj (t)yj (t). Again, this definition is independent of the orientation or the labeling of the edges. If y ∈ D satisfies Ly = λy we will call it a solution of Ly = λy signifying that both the differential equations and the interface conditions are satisfied. Functions solving the differential equations form a 2r-dimensional vector space. Since there are 2r−n0 interface conditions it is reasonable to expect that the space of solutions of Ly = λy is n0 -dimensional. This fact will be proved below. Denote by cj (λ, ·) and sj (λ, ·) the basis of solutions of −yj′′ + qj yj = λyj defined by initial conditions cj (λ, 0) = s′j (λ, 0) = 1 and c′j (λ, 0) = sj (λ, 0) = 0. We collect these functions in the r×r-matrices C(λ, t) = diag(c1 (λ, t), . . . , cr (λ, t)) and S(λ, t) = diag(s1 (λ, t), . . . , sr (λ, t)). A function y satisfying the differential equations may now be expressed as y = (C(λ, ·), S(λ, ·))ξ for an appropriate ξ ∈ C2r . In particular, the function y = (C(λ, ·), S(λ, ·))ξ satisfies the interface conditions (and hence is a solution of Ly = λy) precisely if ξ is in the kernel of the (2r − n0 ) × 2r-matrix J(λ) = I(C(λ, ·), S(λ, ·)). 2.4. Initial value problems Initial value problems do, in general, not have unique solutions on trees. This causes the main differences in the treatment of inverse problems on trees when compared to intervals. However, it is still useful to investigate the set of all solutions for the initial value problem. One of the boundary vertices is being singled out as the “initial point”. Thus, it is now convenient to treat the tree as a rooted tree. Lemma 2.1. The initial value problem Ly = λy, y0 (0) = a, and y0′ (0) = b has a solution for any choice of a, b ∈ C. Proof. This is obvious when the tree has only one internal vertex and follows by induction over the number of internal vertices for general trees. In fact, if there are k limbs attached to v1 (i.e., k + 1 edges), then there are k subtrees for which we know the existence of solutions of the initial value problem by induction hypothesis. For subtree j, where 1 ≤ j ≤ k, we use initial conditions aj = ac0 (λ, 1) + bs0 (λ, 1) and some value for bj . These provide then for a solution on the full tree provided that b1 + · · · + bk = ac′0 (λ, 1) + bs′0 (λ, 1).

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Corollary 2.2. Suppose T has n0 boundary vertices. The vector space of solutions of Ly = λy satisfying homogeneous initial conditions y0 (0) = 0 and y0′ (0) = 0 has dimension n0 − 2.

Proof. Again, this holds obviously for a tree with one internal vertex. Suppose it is true for trees with less than ℓ internal vertices and consider a tree with exactly ℓ internal vertices. Suppose the tree has k limbs. On each of the k subtrees which start at the end of the trunk there is a solution of the initial value problem for function value 0 and derivative 1. These give rise to k − 1 linearly independent solutions of the homogeneous initial value problem on the full tree. Now, given any solution of the homogeneous initial value problem, one can subtract a suitable linear combination of the k −1 basis functions just constructed to obtain a solution on the full tree which is identically zero on the trunk and on the k limbs. It is thus a linear combination of all possible basis functions for each of the homogeneous initial value problems on the subtrees. If subtree j has nj + 1 boundary vertices then we have, by the induction hypothesis, nj − 1 such basis functions. Since k n0 = 1 + j=1 nj we get for the total number of basis functions on the full tree k k − 1 + j=1 (nj − 1) = n0 − 2. These observations give us now immediately the following theorem.

Theorem 2.3. Suppose T has r edges and n0 boundary vertices. Then the vector space of solutions of Ly = λy has dimension n0 . Moreover, the matrix J(λ) has full rank 2r − n0 for every complex λ.

Proof. There are linearly independent solutions which are equal to c0 (λ, ·) and s0 (λ, ·), respectively, when restricted to the trunk of the tree. Let y be any solution and subtract a suitable combination of the functions just mentioned so that the resulting function satisfies homogeneous initial conditions. That function may expressed as a linear combination of the n0 − 2 basis functions constructed in the previous corollary. The last statement follows now from the fundamental theorem of linear algebra. 2.5. The Dirichlet boundary value problem Given a vector f = (f1 , . . . , fn0 )⊤ in Cn0 we are looking for a solution of Ly = λy satisfying the nonhomogeneous Dirichlet boundary conditions y(vj ) = fj when v1 , . . . , vn0 denote the boundary vertices. These conditions are expressed as Ay = f where A = A0 E0 and A0 = (In0 ×n0 , 0n0 ×(r−n0 ) ) assuming that the boundary edges are oriented such that vj = ǫj (0) for j = 1, . . . , n0 . Here In×n and 0n×m denote the identity matrix and the zero matrix of the given dimensions, respectively. We might subsequently drop the subscripts if the dimensions are clear from the context. We also note that the n0 × 2r-matrix A(C(λ, ·), S(λ, ·)) = (In0 ×n0 , 0n0 ×(2r−n0 ) ). Finally, we introduce the 2r × 2r matrix

I 0 A (C(λ, ·), S(λ, ·)) = M (λ) = R(λ) P (λ) I


37

where R represents the first n0 columns and P the last 2r − n0 columns of the matrix J introduced in Section 2.3. Thus solutions of the equation M (λ)ξ = (f, 0)⊤ provide solutions of Ly = λy satisfying the given boundary conditions. The entries of M are entire functions. Therefore the zeros of the determinant of M are isolated unless it is identically equal to zero. This latter possibility can be ruled out most easily under our assumption of a real potential since the zeros of the determinant are eigenvalues of the Dirichlet operator LD , the restriction of L to the set of all y ∈ D which satisfy homogeneous Dirichlet conditions at the boundary vertices. An integration by parts, using the interface and boundary conditions, shows, that the operator LD is self-adjoint so that its eigenvalues are real. Thus, the equation Ly = λy has a unique solution satisfying given (possibly nonhomogeneous) Dirichlet boundary conditions unless λ is one of the countably many real eigenvalues of LD . The geometric multiplicity of a Dirichlet eigenvalue is strictly less than n0 for there is at least one solution of Ly = λy which does not satisfy homogeneous Dirichlet conditions. Since the problem is self-adjoint the geometric multiplicity of a Dirichlet eigenvalue equals its algebraic multiplicity. The algebraic multiplicity in turn equals the order of the eigenvalue as a zero of det M . This may be seen by a generalization of Naimark’s argument for a differential equation with scalar coefficients to one with matrix coefficients.

3. Weyl solutions A solution ψ of Ly = λy is called a Weyl solution if ψ assumes the value one at precisely one of the boundary vertices and the value zero at each of the others. With one of the boundary vertices singled out it is convenient, in this section, to treat the tree as a rooted tree (see Section 2.1). If x and x′ are two points in T we denote their distance by d(x, x′ ) (recall that T is a metric space). The number h = max{d(x, v0 ) : x ∈ T } is called the height of the tree with respect to the root v0 . (The height of a tree depends on which boundary vertex is designated as root). Lemma 3.1. Let ψ(λ, ·) be the Weyl solution for a tree T which assumes the value one at the root and the value zero at each of the branch tips. Then √ ψ0′ (λ, 0) = − −λ + o(1)

as λ tends to infinity on the negative real axis. Furthermore, if s is the label of a boundary edge leading to branch tip ǫs (1), then √ √ ψs′ (λ, 1) = −2 −λ exp(−d(v0 , ǫs (1)) −λ)(1 + o(1))

again as λ tends to infinity on the negative real axis.

Proof. The proof is by induction on the height of the tree. If the tree has height one, then ψ0′ (λ, 0) = m(λ) is the Titchmarsh-Weyl m-function for the problem √ (recall that ψ0 (λ, 0) = 1). It is well known that m(λ) = iz + o(1) as z = i −λ

38


tends to infinity on the positive imaginary axis. Also for the branch tip ǫ0 (1) we have ψ0′ (λ, 1) = −1/s0 (λ, 1) = 2izeiz + O(eiz + ze3iz ). Next assume the validity of our claim for all trees of height at most n and let T be a tree of height n + 1. Let v1 = ǫ0 (1). Then v1 is the root of several, say k, subtrees, whose height is at most n. These will be labeled T1 , . . . , Tk and their trunks, viewed as limbs of the full tree, are correspondingly labeled ǫ1 , . . . , ǫk . The Weyl solution for the tree Tj is denoted by ψ(j, λ, ·). Note that the Weyl solution ψ(λ, ·) for the full tree, restricted to Tj , is a multiple of ψ(j, λ, ·), in fact ψσ (λ, ·) = ψ(λ, v1 )ψs (j, λ, ·)

(3.1)

when s denotes the label of an edge of Tj and σ denotes the label of the same edge when viewed as an edge of T . In particular, ψj (λ, ·) = ψ(λ, v1 )ψ0 (j, λ, ·).

Now we will determine ψ0 (λ, ·), the Weyl solution for T on its trunk, by solving a Riccati equation. To this end we define ψ ′ (λ, ·) . µ(λ, ·) = 0 ψ0 (λ, ·)

Then µ(λ, ·) satisfies the differential equation

µ′ (λ, x) + µ(λ, x)2 + λ − q0 (x) = 0

and the initial condition k

j=1

µ(λ, 1) =

ψj′ (λ, 0)

ψ(λ, v1 )

=

k

ψ0′ (j, λ, 0) = ikz + o(1).

j=1

This problem will be solved by emulating Bennewitz’s approach in [4]. The first step is to find the solution µ0 (λ, ·) for vanishing q0 but with the correct initial condition. Thus 2iz(µ(λ, 1) − iz) µ0 (λ, x) = iz + . (iz + µ(λ, 1)) exp(2iz(x − 1)) + iz − µ(λ, 1) Now recall that q0 ∈ L1 ([0, 1]) and define 1

t e2 x μ0 (λ,u)du q0 (t)dt µ1 (λ, x) = − x

and

µn (λ, x) =

1

x

Then

e2

t n−1 x

j=0

μj (λ,u)du

µn−1 (λ, t)2 dt,

µ′1 (λ, x)

= q0 (x) − 2µ0 (λ, x)µ1 (λ, x),

µ′n (λ, x)

= −µn−1 (λ, x)2 − 2µn (λ, x)

and µn (λ, 1) = 0 for all n ∈ N.

n−1 j=0

µj (λ, x),

n = 2, 3, . . . .

n = 2, 3, . . . ,


39

∞ ∞ ′ Thus, assuming uniform convergence ∞ of n=0 µn and n=0 µn , we have (after some algebra) that the series n=0 µn (λ, ·) satisfies the same initial value problem (the Riccati equation and the initial condition at x = 1) as µ(λ, ·) and hence is equal to it, i.e., ∞ µ(λ, x) = µn (λ, x). (3.2) n=0

In order to investigate convergence we first realize that µ0 (λ, x) = 1 + o(1/z) iz

for k = 1 and µ0 (λ, x) 2e2iz(1−x) (k − 1) =1+ (1 + o(1/z)) iz k + 1 − (k − 1)e2iz(1−x)

for k > 1. Since iz is negative we may estimate in either case that Re(µ0 (λ, x)) ≤ 3iz/4 = −3 Im(z)/4 for sufficiently large z. Thus |µ1 (λ, x)| ≤

1

x

e−3 Im(z)(t−x)/2 |q0 (t)|dt.

Given ε > 0, there are complex numbers α1 , . . . , αN and intervals I1 , . . . , IN such N that q˜0 = j=1 αj χIj is a step function on [0, 1] with q0 − q˜0 ≤ ε. Then |µ1 (λ, x)| ≤ ε +

N j=1

|αj |

1

x

e−3 Im(z)(t−x)/2 χIj (t)dt ≤ ε +

N 2|αj | . 3 Im(z) j=1

This estimate holds regardless of x and proves that |µ1 (λ, x)| tends to zero uniformly in x as Im(z) tends to infinity. Now let a(λ, x) = sup{|µ1 (λ, t)| : x ≤ t ≤ 1}

and assume that z is large enough so that Im(z) ≥ 1 and a(λ, x)/ Im z ≤ 1/2. One then shows by induction that

2n−1 a(λ, x) |µn (λ, x)| ≤ Im(z) Im(z)

using that a(λ, ·) is non-increasing and that j−1

j n−1 n−1 ∞ a(λ, x) 2 a(λ, x) |µj (λ, x)| ≤ Im(z) ≤ 2a(λ, 0). ≤ Im(z) Im(z) Im(z) j=1 j=1 j=1

∞ These estimates show also that n=0 µn (λ, ·) may be differentiated term by term, thus proving the validity of equation (3.2).

40

B.M. Brown and R. Weikard The Weyl solution on the trunk of T is now given as

x ψ0 (λ, x) = exp µ(λ, t)dt . 0

In particular,

ψ0′ (λ, 0) = µ(λ, 0) = iz + o(1) and

ψ(λ, v1 ) = ψ0 (λ, 1) = exp

1

µ(λ, t)dt

0

(3.3)

= eiz (1 + o(1)).

Finally, consider a boundary edge s on tree Tj , different from the trunk of Tj , and denote the distance of the corresponding boundary vertex from the root of Tj by N . Then, according to our induction hypothesis, ψs′ (j, λ, 1) = 2izeizN (1 + o(1)). This edge is also a boundary edge of T with label σ, say. Using (3.1) we find ψσ′ (λ, 1) = ψ(λ, v1 )ψs′ (j, λ, 1) = 2izeiz(N +1) (1 + o(1)).

This and equation (3.3) complete our proof.

4. Green’s function In Section 2.5 we introduced the matrix M (λ) to describe both boundary and interface conditions. Now we introduce also the operators I0 = E0 E0 + D0 D0 and A0 = A0 E0 as well as the matrix A0 (C(λ, ·), S(λ, ·)). M0 (λ) = I0

The solution of the nonhomogeneous system of equations (L − λ)u = h where h ∈ L2 (T ) and where u is subject to the homogeneous boundary conditions Au = 0 as well as the interface conditions Iu = 0 is given by (see [6]) 1 u(t) = Γ(λ, t, t′ )h(t′ )dt′ 0

where

Γ(λ, t, t′ ) = (C(λ, t), S(λ, t)) (M (λ)−1 M0 (λ) − H(t′ − t))

S(λ, t′ ) −C(λ, t′ )

with H being the Heaviside function, i.e., H(t) equals zero or one depending on whether t is negative or positive. We remark that Γ(λ, ·, ·) represents a scalar function G(λ, ·, ·) on T × T via G(λ, x, y) = Γj,k (t, t′ ) when x = ǫj (t) and y = ǫk (t′ ). Lemma 4.1. The function Γ has the following properties: 1. For any t, t′ ∈ [0, 1] the function Γ(·, t, t′ ) is analytic away from the eigenvalues associated with the Dirichlet problem for T .


41

2. If λ is not an eigenvalue associated with the Dirichlet problem for T then the function Γ(λ, ·, ·) has the following properties: (a) Γ(λ, ·, ·) is continuous on [0, 1]2 . (b) Γ(0,0,1) (λ, ·, ·) is continuous in either of the triangles {(t, t′ ) ∈ [0, 1]2 : t < t′ } and {(t, t′ ) ∈ [0, 1]2 : t > t′ }. On the line t = t′ there is a jump discontinuity: lim(Γ(0,0,1) (λ, t, t + τ ) − Γ(0,0,1) (λ, t, t − τ )) = −I. τ ↓0

(c) For fixed t′ ∈ [0, 1] the function Γ(λ, ·, t′ ) satisfies the equation −Γ(λ, ·, t′ )′′ + (Q − λ)Γ(λ, ·, t′ ) = 0

′

on (0, t ) as well as (t′ , 1). (d) For fixed t ∈ [0, 1] the function Γ(λ, x, ·) satisfies the equation −Γ(λ, t, ·)′′ + Γ(λ, t, ·)(Q − λ) = 0

on (0, t) as well as (t, 1). (e) If k ≤ n0 then Γk,j (λ, 0, t′ ) = 0. (f) If j ≤ n0 then Γk,j (λ, t, 0) = 0. Proof. Statements (a) through (d) follow easily from the properties of the Heaviside function and the functions cj and sj . To see the validity of statement (e) note first that M and M0 are both of the form

In0 ×n0 0n0 ×(2r−n0 ) ∗ ∗∗ where ∗ indicates an appropriate (2r − n0 ) × n0 -matrix and ∗∗ indicates an appropriate (2r − n0 ) × (2r − n0 )-matrix. This implies that M −1 M0 is also of that same form. Hence, when t′ > 0, the first n0 rows of M −1 M0 − H(t′ ) are all zero proving (e) in that case. By continuity this is also true when t′ = 0. To prove (f) note that, if t > 0, we have H(−t) = 0. For this case we need to know more about the structure of M0 . Since the bottom 2r − n0 rows contain only information about interior vertices we know that the entries in the first n0 columns and the last 2r − n0 rows of (A0 , I0 )⊤ are zero. Hence M0 has the form

0n0 ×(r−n0 ) 0n0 ×n0 0n0 ×(r−n0 ) In0 ×n0 0(2r−n0 )×n0 ∗ 0(2r−n0 )×n0 ∗∗

where ∗ and ∗∗ indicate appropriate (2r − n0 ) × (r − n0 )-matrices. Thus

Se (λ, t′ ) 0n0 ×(r−n0 ) S(λ, t′ ) = M0 0(2r−n0 )×n0 ∗∗∗ −C(λ, t′ ) where ∗∗∗ indicates a (2r−n0 )×(r−n0 )-matrix and where Se (λ, t′ ) = diag(s1 (λ, t′ ), . . . , sn0 (λ, t′ )). The first n0 columns, corresponding to the requirement j ≤ n0 are zero when t′ = 0.

42


We now express the Weyl solutions in terms of Green’s function. Let h be a function in D, the domain of L satisfying the boundary conditions Ah = eℓ . Next let y be the unique solution of the following inhomogeneous problem (L − λ)y = (λ − L)h,

y ∈ D,

Ay = 0.

Then ψ(ℓ, λ, t) = (y + h)(t). We proceed to compute y. By partial integration 1 1 Γ(λ, t, t′ )h′′ (t′ )dt′ Γ(λ, t, t′ )(λ − Q(t′ ))h(t′ )dt′ + y (t) = 0

=− +

t

Γ

0 1

(0,0,2)

(λ, t, t )h(t )dt − ′

′

′

0

1

Γ(0,0,2) (λ, t, t′ )h(t′ )dt′

t

Γ(λ, t, t′ )h′′ (t′ )dt′

0 (0,0,1)

= −Γ

(λ, t, ·)h|t0 − Γ(0,0,1) (λ, t, ·)h|1t + Γ(λ, t, ·)h′ |t0 + Γ(λ, t, ·)h′ |1t .

In particular, for hℓ (t′ ) = (1 − t′ )2 and hj (t′ ) = 0 for j = ℓ we get y(t) = lim [Γ(0,0,1) (λ, t, t + τ )h(t + τ ) − Γ(0,0,1) (λ, t, t − τ )h(t − τ )] τ →0

+ Γ(0,0,1) (λ, t, 0)h(0) − Γ(λ, t, 0)h′ (0). Since h(0) = eℓ , since column ℓ of Γ(λ, t, 0) is a zero column, and since the term in brackets tends to −h(t) we obtain (0,0,1)

ψj (k, λ, t) = Γj,k

(λ, t, 0).

(4.1)

4.1. Eigenfunction expansion of Green’s function Let LD be the Dirichlet operator introduced in Section 2.5. The eigenvalues λk of LD may have geometric multiplicity larger than one. We will label them in such a way that they are repeated according to their multiplicity and such that λ1 ≤ λ2 ≤ · · · . The corresponding orthonormalized eigenfunctions are denoted by ϕ1 , ϕ2 etc. Then (see Coddington and Levinson [8], pp. 298/299 for a similar situation) ∞ ϕn (x)ϕn (y) G(λ, x, y) = λn − λ n=1 or, equivalently,

Γj,ℓ (λ, t, s) =

∞ ϕn;j (t)ϕn;ℓ (s) . λn − λ n=1

Integration by parts shows that r 1 r 1 |ϕn |2 = (−ϕ′′n;j + qj ϕn;j )ϕn;j = (|ϕ′n;j |2 + qj |ϕn;j |2 ) λn = λn T

j=1

0

j=1

0


43

since rj=1 (ϕ′n;j ϕn;j )|10 = 0 due to the boundary and interface conditions satisfied by ϕn . Because q is real and bounded by a constant C this gives r 1 |ϕ′n;j |2 ≤ C + λn . j=1

Also

r j=1

0

1

|ϕ′′n;j |2 ≤

0

r j=1

0

1

|(qj − λn )ϕn;j |2 ≤ (C + |λn |)2

again using the boundedness of q. The elementary estimate 1 (|f |2 + |f ′ |2 ), |f (x)|2 ≤ 4 0

which holds for an absolutely continuous function defined on [0, 1], shows then the existence of a constant C ′ such that |ϕ′n;j (t)| ≤ C ′ |λn | ∞ ′ −m−1 ′ for all sufficiently large n. Thus the series is n=1 ϕn;j (t)ϕn;ℓ (s)(λn − λ) absolutely and uniformly convergent for m ≥ 2, since the λn , as zeros of an entire ∞ function of growth order 1/2 satisfy n=1 |λn |−1 < ∞. Therefore (m,1,1)

Γj,ℓ

(λ, t, s) = m!

∞ ϕ′n;j (t)ϕ′n;ℓ (s)

n=1

if m ≥ 2.

(λn − λ)m+1

(4.2)

5. The Dirichlet to Neumann Map Recall that there is a unique solution of Ly = λy satisfying the nonhomogeneous Dirichlet boundary conditions yj (0) = fj for j = 1, . . . , n0 unless λ is one of the isolated Dirichlet eigenvalues. One may then compute the values gj = −yj′ (0), j = 1, . . . , n0 , (Neumann data). The relationship between the fj and the gj is linear and is called the Dirichlet-to-Neumann map. We denote this map by Λ. Recall that ψ(k, λ, ·) denotes the Weyl solution which assumes the value one at the boundary vertex vk (and zero at every other boundary vertex). Thus we have Λj,k = −ψj′ (k, λ, 0) and, using equation (4.1), (0,1,1)

Λj,k = −Γj,k

(λ, 0, 0).

(5.1)

In [6] the following theorem was proved. Theorem 5.1. Let q be a complex-valued integrable potential supported on a finite metric tree whose edge lengths are one. Then the associated Dirichlet-to-Neumann map uniquely determines the potential almost everywhere on the tree.

44


This theorem is the analogue for trees of a result of Nachman, Sylvester, and Uhlmann [19] who consider a Schr¨ odinger equation on a bounded domain in Rn . Our proof, given in [6], proceeds in two steps. First, we show that the diagonal elements of the Dirichlet-to-Neumann map determine the potential on the boundary edges. Then we show that the Dirichlet-to-Neumann map of the tree determines the Dirichlet-to-Neumann map for a smaller tree with some of the boundary edges pruned off. As the case of a finite interval can be considered a tree with two boundary vertices, it is clear that not all the information provided by the Dirichlet-to-Neumann map is necessarily needed. In that case it is sufficient to know one of the diagonal entries. Indeed, as Yurko proves in [22], this generalizes to the present case: knowing n0 − 1 of the diagonal elements of the Dirichlet-to-Neumann map is already sufficient to determine the potential almost everywhere on the tree. Yurko shows this by a more careful investigation of the tree pruning procedure than we had performed.

6. A Levinson-type theorem In this section we prove Theorem 1.1. The proof is divided in two lemmas following the idea of Nachman, Sylvester, and Uhlmann [19]. Assume two potentials q and q˜ are given both satisfying the hypotheses of the theorem, that is to be real-valued and bounded. To each is associated a Dirichlet-to-Neumann map, denoted by Λ ˜ respectively. Both problems have the same Dirichlet eigenvalues and the and Λ, same Neumann data for an orthonormal set of Dirichlet eigenfunctions. We prove ˜ is a polynomial and in Lemma in Lemma 6.1 that, under these circumstances, Λ− Λ 6.2 that this polynomial must be zero. The conclusion of the theorem follows then by applying Theorem 5.1, i.e., that the Dirichlet-to-Neumann map determines the potential almost everywhere on the tree. ˜ is a polynomial of degree at most one. Lemma 6.1. Λ − Λ (0,1,1)

Proof. By equation (5.1) we have Λj,k (λ) = −Γj,k equation (4.2), (m)

(m,1,1)

Λj,k (λ) = −Γj,k

(λ, 0, 0) = −m!

(λ, 0, 0). Hence, employing

∞ ϕ′n;j (0)ϕ′n;k (0)

n=1

(λn − λ)m+1

provided m ≥ 2. The right-hand side is determined by the information provided, i.e., the Dirichlet eigenvalues and the Neumann data of the Dirichlet eigenfunc˜ (m) . tions. Therefore the exact same expression is obtained for Λ j,k ˜ tends to zero. Lemma 6.2. As λ tends to negative infinity Λ − Λ

Proof. By Lemma 3.1 the quantities Λj,k = −ψj′ (k, λ, 0) are exponentially small √ except for j = k in√which case we have Λk,k = ψk′ (k, λ, 0) = − −λ + o(1). But ˜ k,k = o(1). ˜ k,k = − −λ + o(1), we have that (Λ − Λ) since also Λ


45

7. A Marchenko-type theorem In this section we prove Theorem 1.2. The proof relies on establishing a relationship between the eigenfunctions associated with a Dirichlet eigenvalue E and the Weyl solutions for λ near E. We denote the set of eigenvalues by Σ, i.e., Σ = {λn : n ∈ N}. If E ∈ Σ we denote its geometric multiplicity by µ(E). Recall that the solution of a Dirichlet boundary value problem is determined by an equation of the form M ξ = f . The characteristic function associated with the operator LD is (a multiple of) the determinant of M , an entire function of growth order 1/2. Hence, by Hadamard’s factorization theorem2 ∞ det(M (λ)) = C (1 − λ/λn ) = C (1 − λ/E)μ(E) n=1

E∈Σ

where C is an appropriate constant. Since our problem is self-adjoint geometric and algebraic multiplicities coincide. In particular, every eigenvalue has index one3 . Therefore we define (1 − λ/E) χ(λ) = E∈Σ

the “minimal function” associated with the operator LD .

Lemma 7.1. Suppose that E is a Dirichlet eigenvalue. Let ξ(j, λ) = (a1 (j, λ), . . . , ar (j, λ), b1 (j, λ), . . . , br (j, λ))⊤ be the unique solution of M (λ)ξ = χ(λ)ej for λ near E and 1 ≤ j ≤ n0 . Then the ξ(j, λ) have a nonzero limit as λ tends to E and the functions (C(E, ·), S(E, ·))ξ(j, E),

j = 1, . . . , n0

span the space of Dirichlet eigenfunctions for E. Proof. We know that µ(E), the multiplicity of the Dirichlet eigenvalue E, is also the multiplicity of E as a zero of det M . Because of the structure of M this means that the lower right (2r−n0 )×(2r−n0 ) block P (E) of M (E) has rank 2r−n0 −µ(E). Without loss of generality, employing elementary row operations, we may assume that the top 2r − n0 − µ(E) rows of P (E) are independent and, consequently, that all entries in the bottom µ(E) rows of P (λ) tend to zero as λ tends to E. This structure is exhibited if we write M in the following way: ⎞ ⎛ I 0 0 ⎝ M = R1 P1 P2 ⎠ R2 P3 P4 where the blocks in the diagonal are of size n0 × n0 , (2r − n0 − µ(E)) × (2r − n0 − µ(E)), and µ(E) × µ(E), respectively. As we just pointed out we know that P3 (λ)

2 We are assuming here that zero is not an eigenvalue. This is without loss of generality since a zero eigenvalue would only change the notation but not the essence of the argument. 3 The index of an eigenvalue is the length of the longest possible Jordan chain.

46


and P4 (λ) tend to zero as λ tends to E. Moreover, R2 (E) has full rank (equal to µ(E)) since, by Theorem 2.3, this is true for

R1 P1 P2 J= R2 P3 P4 regardless of λ. We now multiply the equation M (λ)ξ = χ(λ)ej from the left with the invertible matrix ⎞ ⎛ I 0 0 P1−1 0⎠ T = ⎝ −P1−1 R1 P3 P1−1 R1 − R2 P3 P1−1 I

(suppressing the dependence on λ wherever it is convenient). Thereby we obtain ⎞ ⎛ I 0 0 −1 ⎠ ξ = χ(λ)T ej . ⎝0 I P1 P2 0 0 P4 − P3 P1−1 P2

Writing ξ = (x, y, z)⊤ with appropriately sized columns x and y, and z, we will first consider the equation (P4 − P3 P1−1 P2 )z = χgj where gj is the jth column of P3 P1−1 R1 − R2 . Since all entries of the matrix P4 − P3 P1−1 P2 tend to zero as λ ˜ j where tends to E we may write (P4 − P3 P1−1 P2 )(λ) = (λ − E)F (λ) and F z = χg χ(λ) ˜ = χ(λ)/(λ − E). Since det(T ) = 1/ det(P1 ) assumes a finite nonzero value at E we obtain that det(T M ) = det(P4 − P3 P1−1 P2 ) has a zero at E of the same order as det M , i.e., µ(E). This implies that det F (E) = 0. Also, since χ(E) ˜ = 0 and gj (E) = 0 we obtain that z(j, λ) tends to a nontrivial vector z(j, E). We now may determine also x(j, E) (which will always be zero) and y(j, E). This proves the first statement of the lemma. The second statement follows from the observation that (P3 P1−1 R1 −R2 )(E) = −R2 (E) has rank µ(E) and hence µ(E) linearly independent columns j1 , . . . , jμ(E) , giving rise to linearly independent vectors ξ(j1 , E), . . . , ξ(jμ(E) , E) which in turn provide µ(E) linearly independent Dirichlet eigenfunctions. Corollary 7.2. The functions ω(k, λ, ·) = χ(λ)ψ(k, λ, ·), k = 1, . . . , n0 are defined for all values of λ ∈ C. If λ is not a Dirichlet eigenvalue these functions are linearly independent and span the space of all solutions of Ly = λy. If λ = E is a Dirichlet eigenvalue they span the space of all solutions of LD y = Ey, i.e., the eigenspace associated with the Dirichlet eigenvalue E. In the following we denote derivatives with respect to the spectral parameter by a dot and (as before) derivatives with respect to the spatial variable with a prime. It is easy to check that uv = (u′ v˙ − uv˙ ′ )′ if u(λ, ·) and v(λ, ·) both satisfy the equation −y ′′ + qy = λy for all λ. Hence we get r 1 (ωj′ (k, λ, ·)ω˙ j (ℓ, λ, ·) − ωj (k, λ, ·)ω˙ j′ (ℓ, λ, ·))0 . ω(k, λ, ·)ω(ℓ, λ, ·) = T

j=1


47

Note that both ω(k, λ, ·) and ω(ℓ, ˙ λ, ·) satisfy the interface conditions. Moreover, the boundary conditions, ωj (k, λ, 0) = χ(λ)δj,k and ω˙ j (ℓ, λ, 0) = χ(λ)δ ˙ j,ℓ also hold. Hence we get ′ ˙ ω(k, λ, ·)ω(ℓ, λ, ·) = χ(λ)ω˙k ′ (ℓ, λ, 0) − χ(λ)ω ℓ (k, λ, 0) T

and, in particular,

T

′ ω(k, E, ·)ω(ℓ, E, ·) = −χ(E)ω ˙ ℓ (k, E, 0).

(7.1)

We introduce the symmetric matrix N (E) by setting Nk,ℓ (E) = ω(k, E, ·)ω(ℓ, E, ·). T

Proof of Theorem 1.2. We will prove this theorem by computing the Neumann data of an orthonormal basis of Dirichlet eigenfunctions from the given data. Let E be a Dirichlet eigenvalue of multiplicity µ(E). Since the naming of the boundary vertices is unimportant we assume now that K(E) = {1, . . . , µ(E)}. We may introduce an orthonormal basis ϕ(k, E, ·), k = 1, . . . , µ(E) of the eigenspace of E by writing ⎛ ⎞ ⎛ ⎞ ω(1, E, ·) ϕ(1, E, ·) ⎜ ⎟ ⎜ ⎟ .. . .. ⎝ ⎠=C⎝ ⎠ . ω(µ(E), E, ·)

ϕ(µ(E), E, ·)

with a lower triangular matrix whose diagonal elements are non-zero. In fact, CC ⊤ is the LU-factorization of the upper left µ(E) × µ(E) block of N (E). Thus, this block determines the matrices C and C −1 uniquely. Now, the Neumann data of the orthonormal Dirichlet eigenfunctions ϕ(k, E, ·) are given as linear combinations (in terms of the matrix C) of the Neumann data of the ω(k, E, ·), k = 1, . . . , µ(E) which in turn are uniquely determined, according to equation (7.1) by the first µ(E) columns of N (E) and the known function χ.

References [1] S. Avdonin and P. Kurasov. Inverse problems for quantum trees. Preprint, 2006. [2] M.I. Belishev. Boundary spectral inverse problem on a class of graphs (trees) by the BC method. Inverse Problems, 20(3):647–672, 2004. [3] M.I. Belishev and A.F. Vakulenko. Inverse problems on graphs: recovering the tree of strings by the BC-method. J. Inverse Ill-Posed Probl., 14(1):29–46, 2006. [4] C. Bennewitz. A note on the Titchmarsh-Weyl m-function. Preprint ANL-87-26, Vol. 2, 105–111. Argonne National Laboratory, Argonne, Illinois, USA, 1988. [5] G. Borg. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math., 78:1–96, 1946. [6] B.M. Brown and R. Weikard. A Borg-Levinson theorem for trees. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461(2062):3231–3243, 2005.

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[7] R. Carlson. Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc., 351(10):4069–4088, 1999. [8] E.A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. Robert E. Krieger, Malabar, Florida, 1984. [9] E.B. Curtis and J.A. Morrow. Determining the resistors in a network. SIAM J. Appl. Math., 50(3):918–930, 1990. [10] E.B. Curtis and J.A. Morrow. The Dirichlet to Neumann map for a resistor network. SIAM J. Appl. Math., 51(4):1011–1029, 1991. [11] N.I. Gerasimenko. The inverse scattering problem on a noncompact graph. Teoret. Mat. Fiz., 75(2):187–200, 1988. [12] M. Harmer. Inverse scattering on matrices with boundary conditions. J. Phys. A, 38(22):4875–4885, 2005. [13] P. Kuchment. Graph models for waves in thin structures. Waves Random Media, 12(4):R1–R24, 2002. [14] P. Kuchment. Quantum graphs. I. Some basic structures. Waves Random Media, 14(1):S107–S128, 2004. Special section on quantum graphs. [15] P. Kurasov and F. Stenberg. On the inverse scattering problem on branching graphs. J. Phys. A, 35(1):101–121, 2002. [16] P. Kurasov and M. Nowaczyk. Inverse spectral problem for quantum graphs. J. Phys. A, 38(22):4901–4915, 2005. [17] N. Levinson. The inverse Sturm-Liouville problem. Mat. Tidsskr. B., 1949:25–30, 1949. [18] V.A. Marˇcenko. Concerning the theory of a differential operator of the second order. Doklady Akad. Nauk SSSR. (N.S.), 72:457–460, 1950. [19] A. Nachman, J. Sylvester, and G. Uhlmann. An n-dimensional Borg-Levinson theorem. Comm. Math. Phys., 115(4):595–605, 1988. [20] V.N. Pivovarchik. Ambartsumyan’s theorem for the Sturm-Liouville boundary value problem on a star-shaped graph. Funktsional. Anal. i Prilozhen., 39(2):78–81, 2005. [21] V.N. Pivovarchik. Inverse problem for the Sturm-Liouville equation on a simple graph. SIAM J. Math. Anal., 32(4):801–819 (electronic), 2000. [22] V. Yurko. Inverse spectral problems for Sturm-Liouville operators on graphs. Inverse Problems, 21(3):1075–1086, 2005. B.M. Brown School of Computer Science, University of Wales Cardiff, PO Box 916 Cardiff CF2 3XF, UK e-mail: [email protected] R. Weikard Department of Mathematics University of Alabama at Birmingham Birmingham Alabama 35294-1170, USA e-mail: [email protected]


Wave Operators for Nonlocal Sturm-Liouville Operators with Trivial Potential E.V. Cheremnikh Abstract. The wave operators for the pair of Sturm-Liouville operators corresponding to different nonlocal boundary conditions are considered. By analogy with an absolutely continuous subspace in the selfadjoint case a so-called essential subspace is defined. This definition is given for a pair of operators (not one operator only). In the case of the same finite sets of spectral singularities of two operators the similarity of corresponding semigroups is proved. The domain of definition of the semigroup has finite codimension. Mathematics Subject Classification (2000). 47B38, 34L25. Keywords. Wave operator, spectral singularity, absolutely continuous subspace.

1. Introduction In [1] the perturbation of a boundary condition within the scattering theory was considered for selfadjoint operators. Among the works concerning dissipative operators we note [2] where Sturm-Liouville operators on a half-line with the usual boundary condition were considered. Here the space which plays the role of an absolutely continuous subspace was defined (in terms of the Hardy space H 2 ) like the construction within the well-known functional model (see, e.g., [3–4]). In the present article we consider similar questions for the Sturm-Liouville operator with trivial potential which is non-selfadjoint because of the nonlocal boundary condition. Apropos, in [5] in the space L2 (0, ∞) the operator −id/dx (i.e., the ∞ operator with trivial potential too) with nonlocal condition u(x)dσ(x) = 0 was 0

studied, here in the first place the point spectrum was investigated. It is necessary to note that local wave operators W± (∆) (∆ an arbitrary interval without singularities) are investigated for much more general non-selfadjoint operators (see, e.g., [6–7]).

50

E.V. Cheremnikh

In [8] where the non-selfadjoint Friedrichs’ model was considered the function of the operator just in presence of spectral singularities was studied. Nonlocal perturbation of a boundary condition was introduced within Friedrichs’ model in [9]. So, one can understand the present article as some development of [9] too. We use the method of calculus which belongs to the direction defined by [6]. Our version is more closed to [10] and in the present article itself we emphasize the use of the Laplace transformation. We note too that the present article is a direct prolongation of [11] where an exponential function is given for the operator indicated in our title.

2. Certain notations and auxiliary statements Let Ly = −y ′′ and D(B) = y ∈ L2 (0, ∞) : Ly ∈ L2 (0, ∞), y(0) = 0 By = Ly, y ∈ D(B).

Let η ∈ L2 (0, ∞) be an arbitrary function, Φ(y) = (y, η)L2 (0,∞) and D(A) = z ∈ L2 (0, ∞) : Lz ∈ L2 (0, ∞), z(0) + Φ(z) = 0 Az = Lz, z ∈ D(A).

(2.1)

(2.2)

We call A a perturbed operator with respect to the operator B. We denote by e ∈ L2 (0, ∞) an auxiliary function such that Le, L2 e ∈ L2 (0, ∞),

e(0) = 1,

About the element η we suppose that ∞ |B 3 η(x)|eεx dx < ∞,

Φ(e) = −2

ε > 0.

(2.3)

(2.4)

0

The element η satisfies the conditions of Lemma 2.4 if, for example η(0) = η ′′ (0) = η IV (0) = 0,

η, η′′ , η IV ∈ L2 (0, ∞)

(2.5)

All statements of Section 2 were proved in [11]. Lemma 2.1. Let the operator IA : L2 (0, ∞) → L2 (0, ∞) be defined by the relation D(IA ) = D(B);

IA y ≡ y + Φ(y)e, y ∈ D(B)

−1 z = z + Φ(z)e, z ∈ D(A). Then IA D(B) = D(A) and IA

We need the Fourier transformation F : L2 (0, ∞) → H ≡ L2ρ (0, ∞), ρ(τ ) = 1√ τ where π √ ∞ sin x τ dx, τ > 0 Fy(τ ) = y(x) √ τ 0

Wave Operators for Nonlocal Sturm-Liouville Operators

51

and we need the element eζ ∈ L2 (0, ∞), ζ ∈ [0, ∞) such that Feζ (τ ) = Let δ(ζ) = 1 + Φ(eζ ) = 1 +

∞ 0

1 , τ −ζ

τ > 0,

Fη(s)ρ(s) ds, s−ζ

ζ ∈ [0, ∞)

ρ(s) =

1√ s, π

(2.6)

ζ ∈ [0, ∞).

(2.7)

Let δ± (σ) = lim δ(σ + iτ ), σ > 0. τ →±0

Lemma 2.2. If the conditions (2.3), (2.4) hold then the function δ(ζ), √ ζ ∈ [0, ∞) (see (2.7)) admits an analytic prolongation δ± (ζ) in the domain |ℑ ζ| < ε and lim (δ± (ζ) − 1) = 0 uniformly in the corresponding domain ℑ ζ > −ε1 or ℑ ζ < |ζ|→∞

ε1 for every ε1 > 0. We denote the resolvent of A by Rζ (A) = (A − ζ)−1 and the resolvent set by ρ(A). There exists a linear subspace L dense in L2 (0, ∞) such that the limit values (Rσ (A)f, g)± = lim (Rσ+iτ (A)f, g)L2 (0,∞) , τ →±0

f, g ∈ L,

σ>0

(2.8)

are smooth functions. A generalized pole of at least one of the function (2.8) is called spectral singularity of the operator A. Theorem 2.3. 1) Let ζ ∈ [0, ∞), then ζ ∈ ρ(A) iff δ(ζ) = 0, in this case Rζ (A)f = Rζ (B)f −

1 (Rζ (B)f, η)L2 (0,∞) eζ δ(ζ)

(2.9)

where the element eζ ∈ L2 (0, ∞) is defined by the relation (2.6). A value ζ ∈ [0, ∞) is an eigenvalue of the operator A iff δ(ζ) = 0. 2) Let σ > 0 and δ± (σ) = 0. Then σ belongs to the continuous spectrum of A. A value σ > 0 is a spectral singularity of the operator A iff δ+ (σ) = 0 or δ− (σ) = 0. 3) The set of eigenvalues and spectral singularities of the operator A is finite. We denote

m(s) = i δ0′ , eisB η ,

n(s) = − y0 + Φ(y0 )e, eisB η L2 (0,∞)

2

where y0 is an element from L (0, ∞) and 3

(δ0′ , f )

(2.10)

≡ f ′ (0).

∗

Lemma 2.4. Let η ∈ D(B ) ∩ D(A ). Let us consider the problem z˙ = −iAz,

z|t=0 = z0 ∈ D(A),

t>0

(2.11)

52

E.V. Cheremnikh

where z(t) is an unknown function and the problem (see (2.10)) y˙ = −iBy + f (t), k(t) +

t 0

y|t=0 = y0 ∈ D(B),

t>0

m(t − s)k(s)ds = n(t)

(2.12) (2.13)

where

˙ f (t) ≡ −k(t)e − ik(t)Le and (y(t), k(t)) are two unknown functions. If z(t) is a solution of problem (2.11) then the functions y(t) = z(t) + Φ(z(t))e,

k(t) = −Φ(z(t))

define the solution of the problem (2.12)–(2.13) where y0 = z0 + Φ(z0 )e. Inversely, if (y(t), k(t)) is a solution of problems (2.12)–(2.13) then z(t) = y(t) + Φ(y(t))e is a solution of problem (2.11) where z0 = y0 + Φ(y0 )e. Let k(·) be some locally integrable functions on (0, ∞) and let t Et± k(τ ) ≡ e±isτ k(s)ds, t, τ > 0.

(2.14)

0

Obviously, if k ∈ C 1 [0, ∞) then Et± k ∈ L2ρ (0, ∞), t > 0. We denote by ∞ I(k, p) ≡ e−sp k(s)ds 0

the Laplace transformation of the function k. The following theorem represents the exponential function of the non-selfadjoint operator A. Theorem 2.5. Suppose that the element η ∈ L2 (0, ∞) satisfies conditions (2.4), (2.5). Then the problem (2.11) is uniformly correct and its solution for t > 0 takes the form (see (2.14)) z(t) = e−itB z0 + ie−itB F −1 Et+ k, t > 0

(2.15)

I(k, p) = i (Rip (A)z0 , η)L2 (0,∞) , ℜp > p0

(2.16)

where the function k(s), s > 0 is defined by the relation and for t < 0 takes the form z(t) = e−itB z0 − ie−itB F −1 Et− k1 ,

t 0 is defined by the relation I(k − , p) = −i (R−ip (A)z0 , η)L2 (0,∞) .

(2.18)

Expression (2.15) represents the exponential function of the operator exp(−itA), t > 0 (see (2.17) in the case t < 0). This explicit form permits later to study corresponding wave operators.


53

3. Wave operators Formally relations (2.15)–(2.17) give immediately the wave operators for the pair A, B or the pair A, A1 where the operator A1 is defined by analogy with operator A. But such wave operators in the general case are unbounded and our aim is to indicate the sufficient conditions of boundedness of these operators. Function k (see (2.15)) is given by its Laplace’s transformation (2.16). In view of equality (2.16) we introduce the following definition. Recall that according to the relation (2.9) we have for λ ∈ ρ(A) (Rλ (A)z, u)L2 (0,∞) = (Rλ (B)z, u)L2 (0,∞) −

(Rλ (B)z, η)L2 (0,∞) (eλ , u)L2 (0,∞) . δ(λ) (3.1)

We denote 1√ τ. π Definition 3.1. Let Hess (A) ⊂ H0 be a linear subspace of all elements z ∈ H0 such that for every element w ∈ H0 the function ζ → (Rζ (A)z, w)H0 , ζ ∈ ρ(A) belongs to the Hardy space in both half-plains ℑ ζ > 0 and ℑ ζ < 0, i.e., H0 = L2 (0, ∞), H = L2ρ (0, ∞), ρ(τ ) =

Hess (A) = {z ∈ H0 : (Rζ (A)z, w)H0 ∈ H 2 (ℑ ζ > 0) ∩ H 2 (ℑ ζ < 0)}

(3.2)

Further we write “belongs to H 2 ” instead “belongs to H 2 (ℑ ζ > 0) and belongs to H 2 (ℑ ζ < 0)”. Let U = {u ∈ H0 : ∃M (u) > 0 : |(eζ , u)H0 | M (u), ζ ∈ [0, ∞)}.

Note that if u ∈ U then

π|(Fu)(τ )|ρ(τ ) M (u),

τ > 0.

(3.3) (3.4)

Therefore (Rζ (B)u, w)H0 ∈ H 2 , w ∈ H0 , u ∈ U, (3.5) i.e., for the non-perturbed operator B we have U ⊂ Hess (B). We can rewrite (3.5) as (3.6) (Rζ (B)z, u)H0 ∈ H 2 , z ∈ H0 , u ∈ U. The analogue relation for the perturbed operator A holds for z from some subspace in H0 only. But this subspace does not depend from an element u ∈ U . Proposition 3.2. Let z ∈ H0 , then

(Rζ (A)z, u)H0 ∈ H 2 , ∀u ∈ U ⇐⇒ (Rζ (A)z, η)H0 ∈ H 2 .

(3.7)

Proof. As the elements η satisfy (2.4) then η ∈ U . So, it remains to consider the direction (⇐=). Let (Rζ (A)z, η)H0 ∈ H 2 . Due to the relation (Rζ (A)z, u)H0 = (Rζ (B)z, u)H0 − (Rζ (A)z, η)H0 (eζ , u)H0

and (3.3), (3.6) too we obtain (Rζ (A)z, u)H0 ∈ H 2 . The proposition is proved.

Note that the subspace Hess (A) ⊂ H0 has a finite codimension, codimHess (A) < ∞.

54

E.V. Cheremnikh Taking into account Proposition 3.2 we introduce

Definition 3.3. Let D(Γ± ) = Hess (A) and (Γ± z)(τ )(Rτ (A)z, η)±

1 (Rτ (B)z, η)± , δ± (τ )

z ∈ Hess (B).

(3.8)

Obviously the operator Γ± acts from Hess in H0 . Proposition 3.4. Let z ∈ D(A) ∩ Hess (A). Then the limit value lim eitB e−itA z ∈ L2 (0, ∞) = H0

t→±∞

exists iff Γ± z ∈ L2 (0, ∞) ∩ L2ρ (0, ∞) = H0 ∩ H. Proof. As z ∈ D(A) ∩ Hess (A) then we have p0 = 0 in relation (2.16) and according to (3.8) ∞

eiλs k(s)ds =

0

i (Rλ (B)z, η)+ = i(Γ+ z)(λ), δ+ (λ)

Consequently k ∈ L2 (0, ∞). As z ∈ D(A) in view of (2.14)–(2.15) we have ⎞ ⎛ t isτ itB −itA −1 ⎝ e e e k(s)ds⎠ , z = z + iF

ℑ λ = 0.

t > 0.

(3.9)

(3.10)

0

Therefore the limit value in (3.10) in the case t → +∞ exists iff the improper integral t ∞ eisτ k(s)ds (3.11) eisτ k(s)ds = lim t→+∞

0

0

converges in the space h = L2ρ (0, ∞). The case t < 0 is considered by analogy. The proposition is proved. Definition 3.5. Let D(W±0 (B, A)) = {z ∈ D(A) ∩ Hess (A) : Γ± z ∈ H0 ∩ H}

(3.12)

W±0 (B, A)z = s − lim eitB e−itA z,

(3.13)

and t→±∞

z ∈ D(W±0 (B, A)),

then the operators W±0 (B, A) : H0 → H0 are called initial wave operators. The closure of the operators W±0 (B, A) in the space H0 , W± (B, A) = W±0 (B, A) are called wave operators.

(3.14)


55

We need too the notation L2ε (0, ∞) = z ∈ H0 : Fz(τ ) = O(τ −2 ), τ → ∞, √ ε . uniformaly for |ℑ τ | 2

(3.15)

Theorem 3.6. If the element η ∈ H0 satisfies conditions (2.4), (2.5) then (see (3.13)) (3.16) W±0 (B, A)z = z − F −1 Γ± z, z ∈ D(W±0 (B, A)) and for every ε > 0

D(A) ∩ D(Γ± ) ∩ L2ε (0, ∞) ⊂ D(W±0 (B, A)).

(3.17) 2

Further we need the following additional condition on the element η ∈ L (0, ∞): the transformation Fη(τ ) is a rational function which is bounded on the half-line (0, ∞). As √ 1 F(ei ζx )(τ ) = , τ > 0, ℑ ζ > 0 τ −ζ then such an element η ∈ H0 is a finite sum η(x) = (3.18) pj (x)eαj x j

where pj (x) is an arbitrary polynomial and αj an arbitrary complex number, ℜ αj < 0. In view of (2.7) and the relation ∞ 0

√ s πi √ , ds = √ (s − aj )(s − ζ) aj + ζ

ℑ

√ aj , ℑ ζ > 0

√ we obtain that δ± (τ ) are rational functions on τ , which are bounded on [0, ∞). We will suppose too that the value δ± (0) is finite and δ± (0) = 0.

(3.19)

Lemma 3.7. Suppose conditions (3.18)–(3.19) hold. If W± (B, A) = W±0 (B, A) (see (3.12), (3.16)) then D(W± (B, A)) = {z ∈ Hess (A) : Γ± z ∈ H0 ∩ H}

(3.20)

W± (B, A)z = z − F −1 Γ± z,

(3.21)

and Corollary 3.8. Hess (A) ∩

L2ε (0, ∞)

z ∈ D(W± (B, A)).

⊂ D(W± (B, A)).

Proof. Condition z ∈ D(A) in Theorem 3.6 serves for the existence itself of the representation (3.16) only. So, we can repeat the proof of Theorem 3.6 for z ∈ Hess (A) ∩ L2ε (0, ∞).

56

E.V. Cheremnikh

4. Properties of wave operators We will make more precise the form of W± (B, A), give the connection with singular integral operators and prove the invertibility of the wave operator. Theorem 4.1. If z ∈ D(W± (B, A)) and ψ = Fz ∈ C 1 (0, ∞) then FW± (B, A)F −1 ψ(τ ) = (Lp,q ψ(τ ) + Rψ(τ ))/δ± (τ ) where ⎧ ∞ ⎪ ⎪ ψ(s) ⎨ L ψ(τ ) = p(τ )ψ(τ ) + q(τ )H[ψ](τ ), H[ψ](τ ) = 1 V.p. ds p,q π s−τ ⎪ 0 ⎪ ⎩ 1 p(τ ) = 12 (δ+ (τ ) + δ− (τ )), q(τ ) = − 2i (δ+ (τ ) − δ− (τ ))

(4.1)

(4.2)

and

Rψ(τ ) =

∞

r(s, τ ) = −Rτ (γρ)(s)

r(s, τ )ψ(s)ds,

0

(4.3)

where γ = Fη, Rτ h(s) ≡ (h(s) − h(τ ))/(s − τ ). Proof. In view of relations (3.8), (3.21) we have FW± (B,A)F −1 ψ(τ ) = ψ(τ ) − Γ± F −1 ψ(τ ) = ψ(τ ) − 1 = ψ(τ ) − δ± (τ )

∞ 0

ψ(s)γ(s) ρ(s)ds s−τ

1 (Rτ (B)F −1 ψ,F −1 γ)± δ± (τ )

±

# $ ∞ ψ(s)γ(s) 1 (δ± (τ ) ∓ πiγ(τ )ρ(τ ))ψ(τ ) − V.p. ρ(s)ds . = δ± (τ ) s−τ 0

Using the equality (see (2.9)) δ± (τ ) = 1 +

∞ 0

γ(s) ρ(s)ds s−τ

±

= 1 ± πiγ(τ )ρ(τ ) + V.p.

∞ 0

γ(s) ρ(s)ds s−τ

and the elementary relation V.p.

∞ 0

ψ(s)γ(s) ρ(s)dsπγ(τ )ρ(τ )H[ψ](τ ) + s−τ

∞ 0

Rτ (γρ)(s)ψ(s)ds

we obtain equalities (4.1)–(4.3). The corollary is proved.


57

Lemma 4.2. Let p, q, a, b be arbitrary functions from the space C 1 [0, ∞). Then the operators Lp,q , La,b (see (4.2)) are bounded in the space H and ∞ Lp,q La,b θ = Lpa−qb,pb+qa θ + qKa,b θ, Ka,b θ(τ ) = ka,b (τ, s)θ(s)ds (4.4) 0

where

1 1 Rτ a(s) − H[Rτ b](s) + b(τ )Rτ ln s π π Note that the proof uses the relation ∞ H2 [θ](τ ) = −θ(τ ) + Rτ ln s θ(s)ds. ka,b (τ, s) =

0

Lemma 4.3. The operator δ± (·)FW± (B, A)F −1 admits a continuous extension as bounded singular integral operator δ+ (τ )FW± (B, A)F −1 ψ(τ ) = Lp, qψ(τ ) + Rψ(τ ),

ψ ∈ F D(W+ (B, A))

(4.5)

on the subspace FD(W+ (B, A)), where D(W+ (B, A)) is a closure in H0 of the domain of definition D(W+ (B, A)). √ Proof. As it was mentioned the function δ± (τ ) is a rational function on τ bounded on [0, ∞) (see (2.9), (3.18)). So, the functions p(τ ), q(τ ) are bounded on [0, ∞) too (see (4.2)). The Hilbert transformation H is bounded in the space L2ρ (0, ∞) where 1√ s. Therefore the operator Lp,q (see (4.2)) is bounded in L2ρ (0, ∞). ρ(s) = π It remains to prove that the integral operator ∞ Rψ(τ ) = − Rτ (γρ)(s)ψ(s)ds, γ = Fη 0

is compact in the space L2ρ (0, ∞). Recall that an operator Lf (τ ) =

∞

l(τ, s)f (s)ρ(s)ds

(4.6)

0

is compact in the space L2ρ (0, ∞) if ∞ ∞ 0

0

(see [13], Ch. 10.2). So, we must prove that ∞ ∞ 0

0

|l(τ, s)|2 ρ(τ )ρ(s)dτ ds < ∞

|Rτ (γρ)(s)|2

ρ(τ ) dτ ds < ∞. ρ(s)

(4.7)

(4.8)

58

E.V. Cheremnikh

1√ s, s > 0 is a linear combination of But the function Rτ (γρ)(s), ρ(s) = π terms under the form

1 1 √ √ √ = √ (4.9) Rτ √ s+a ( s + a)( s + τ )( τ + a) and their derivatives with respect to a, what proves (4.8). The lemma is proved.

Proposition 4.4. The “jump” of the resolvent of the operator A on the continuous spectrum has the form (Rσ (A)u, v)+ − (Rσ (A)u, v)− = (u, bσ )(aσ , v),

where (see (2.7))

δ± = 0

⎧ ⎨ (a , v) = 1 (e , v) − 1 (e , v) σ + σ − σ δ+ (σ) δ− (σ) ⎩ (u, bσ ) = δ− (σ)Fu(σ) − (Rσ (B)u, η)−

(4.10)

(4.11)

for Fu, Fv ∈ C 1 (0, ∞). If in addition u ∈ D(W+ (B, A)) ∩ D(W− (B, A)), then (u, bσ )δ+ (σ)W+ (B, A)u(σ)δ− (σ)W− (B, A)u(σ).

(4.12)

Corollary 4.5. The operators W± (B, A) are invertible. Really, if W± (B, A)u = 0 then “jump” is zero, therefore u = 0.

5. Similarity of semigroups In this section we will discuss the similarity of two semigroups exp(itA) and exp(itA1 ) for the operators A and A1 with the same set of spectral singularities. We will use the notation S(A) = {σj , kj± }

(5.1)

where {σj } denotes the set of all spectral singularities of the operator A and kj± denotes the multiplicity σj as root of the function δ± (σ), σ > 0. Let A1 be an other operator defined like the operator A (see (2.3)) by an element η1 ∈ L2 (0, ∞) under the form (3.18) and let δ1 (ζ) = 1 + (eζ , η1 )L2 (0,∞) , ζ ∈ [0, ∞) (compare with (2.81)). We introduce the following notation: S(τ ) =

δ− (τ ) , δ+ (τ )

S1 (τ ) =

and

δ1,− (τ ) , δ1,+ (τ )

∆+ (τ ) =

δ+ (τ ) δ1,+ (τ )

(5.2)

S1 (τ ) 1 S1 (τ ) 1 1+ , n(τ ) = − 1− , τ > 0. (5.3) 2 S(τ ) 2i S(τ ) kj We need some additional notation too. Let P (s) = (s − σj ) , n = kj be some polynomials and Q(s) = P (s)/(1 + s)n . Using the coefficients of the decomposition 1 pjk , pjk = const =1+ Q(s) (s − σj )k m(τ ) =

j,k

Wave Operators for Nonlocal Sturm-Liouville Operators we introduce the functional D[F (σ)] = j,k

pjk F (k−1) (σj ), (k − 1)!

59

F ∈ C n (0, ∞),

the functions

Qlj (s) =

kj 1 Q(s) pjk (s − σj )l−k , l! k=l+1

l = 0, 1, . . . , kj − 1

and the operator πS : C n (0, ∞) → C n (0, ∞) πS h(s) =

j −1 k

j

h(l) (σj )Qlj (s).

l=0

An elementary calculation results in the following proposition. 2 Proposition 5.1. The operator πS is the projection, πS = πS , and

h(s) = πS h(s) + Q(s)Rh(s),

h ∈ C n (0, ∞)

(5.4)

where Rh(s) = h(s) + D[Rσ h](s),

Rσ h(s) = (h(s) − h(σ))/(s − σ).

Note that the decomposition (5.4) is unique in the following sense: the equalities h(s) = h1 (s) + Q(s)h2 (s), and h1 = πS h, are equivalent.

h1 ∈ R(πS )

(5.5)

h2 = Rh

(5.6)

Definition 5.2. Let X± (A, A1 ) : Hess (A1 ) → Hess (A) be the operator with the domain of definition D(X± (A, A1 )) = {z ∈ Hess (A1 ) : there exists w ∈ Hess (A) (5.7) such that W± (B, A)w = W± (B, A1 )z} and the action X± (A, A1 ) = W± (B, A)−1 W± (B, A1 ).

(5.8)

Recall that the operators W± (B, A), W± (B, A1 ) are invertible, so X± (A, A1 )−1 = X± (A1 , A).

(5.9) 2

Definition 5.3. The closure of the operators X± (A, A1 ) in the space L (0, ∞) is called wave operator : (5.10) W± (A, A1 ) = X(A, A1 ). Operator A is called regular if the function Fη(τ ) is rational. Instead of Γ± we will write Γ± (A) or Γ± (A1 ).

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E.V. Cheremnikh

Theorem 5.4. Let A, A1 be regular operators and S(A) = S(A1 ) (see (5.1)), then 1) the operators W± (A, A1 ), W± (A, A1 )−1 are bounded and W± (A, A1 )−1 = W± (A1 , A);

(5.11)

2) the subspaces M± = D(W± (A, A1 )),

N± = R(W± (A, A1 ))

are closed subspaces respectively in Hess (A1 ) and Hess (A) and have a finite codimension. Proof. Let us consider the operator W+ (A, A1 ) only. In view of (3.21) the equation W+ (B, A)w = W+ (B, A1 )z,

Γ+ (A1 )z ∈ H

for an unknown element w is equivalent to the equation w − F −1 Γ+ (A)w = z − F −1 Γ+ (A1 )z,

Γ+ (A1 )z ∈ H.

(5.12)

Note that the equation under the form Γ+ (A)w = f,

f ∈H

is equivalent to a system containing a singular integral equation for the function ψ = Fw ⎧ 1 ⎨ (Rτ (B)w, η)+ = f (τ ), f ∈ H δ+ (τ ) ⎩ (w, wk ) = 0, k = 1, . . . , k0

where {wk } denotes the set of root elements and functionals corresponding to all eigenvalues and spectral singularities of the operator A (see Definition 3.1). The expression (w, wk ) signifies the scalar product or value of the functional wk . Consequently, if ϕ = Fz, ψ = Fw then equation (5.12) is equivalent to the system (see (4.1)–(4.3)) ⎧ 1 1 ⎨ [Lp,q ψ(τ ) + Rψ(τ )] = [Lp1 ,q1 ϕ(τ ) + R1 ϕ(τ )] δ+ (τ ) δ1,+ (τ ) ⎩ (ψ, ψk ) = 0, k = 1, . . . , k0 where ψk = Fwk . Let (see (5.2))

p0 (τ ) = ∆+ (τ )p1 (τ ),

q0 (τ ) = ∆+ (τ )q1 (τ ),

R0 ϕ(τ ) = ∆+ (τ )R1 ϕ(τ ).

Then the considered system takes the form Lp,q ψ + Rψ = Lp0 ,q0 ϕ + R0 ϕ (ψ, ψk ) = 0,

k = 1, . . . , k0

(5.13) (5.14)

We are looking for a solution of the form ψ = Lm,n ϕ + La,b θ,

θ∈H

(5.15)


61

where (see (5.3)) a=p=

1 (δ+ + δ− ), 2

b = −q =

1 (δ+ − δ− ), 2i

and we denote Ha,b = {ϕ ∈ H : the solution of (5.13) has the form (5.15)}.

(5.16)

Substituting (5.15) into (5.13) and taking into account the relations pm−qn = p0 , qm + pn = q0 and also pa − qb = p2 + q 2 = δ+ δ− , pb + qa = 0 we obtain the equation (see Lemma 4.2) ∞ δ+ (τ )δ− (τ )θ(τ ) + M (s, τ )θ(s)ds = f (τ ) (5.17) 0

where the kernel M (s, τ ) is defined by the identity ∞ −b(τ )Ka,b θ(τ ) + RLa,b θ(τ ) = M (s, τ )θ(s)ds.

(5.18)

0

The right side of the equation (5.17), i.e., f (τ ) = R0 ϕ(τ ) − RLm,n ϕ(τ ) − q(τ )Km,n ϕ(τ ),

(5.19)

is an integral operator on ϕ:

f (τ ) ≡

∞

F (s, τ )ϕ(s)ds.

(5.20)

0

Let σj , j = 1, . . . , j0 be roots of the function δ+ (σ)δ− (σ) of multiplicity kj and P (σ), Q(σ) corresponding functions in Definition 5.1. In view of condition (3.19) the function ξ(τ ), defined by the relation δ+ (σ)δ− (σ) = Q(τ )ξ(τ ), is bounded, i.e., C1 |ξ(τ )| C2 , τ > 0 for some C1 , C2 > 0. Using relations (5.4)–(5.6) we obtain that equation (5.17) is equivalent to the system ∞ ξ(τ )θ(τ ) + (RM (s, ·))(τ )θ(s)ds = Rf (τ ), τ > 0, (5.21) 0

∞ (πS M (s, ·))(τ )θ(s)ds = πS f (τ ).

(5.22)

0

The integral operator in relation (5.21) is compact in view of the next Lemma 5.5 (recall that Fη(τ ), Fη1 (τ ) are rational functions). So, the solution θ(τ ) of equation (5.21) exists iff Rf ⊥ g1 , . . . , qα ,

gi ∈ H

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E.V. Cheremnikh

where {gi } is a finite set of elements. According to (5.19)–(5.20) the last condition signifies that ϕ ⊥ ϕ1 , . . . , ϕα ϕi ∈ H. (5.23) The solution θ of (5.21) as operator on ϕ represents a bounded operator in H. Consequently, the space Ha,b (see (5.16)) is closed and, in view of (5.23), codimHa,b < ∞.

(5.24)

The codimension of the domain of values La,b θ is finite too. So, under the form (5.15)–(5.16) we describe all solutions ψ except maybe a subspace of finite dimension. Now we come back to the system (5.13)–(5.14). Let equation (5.13) hold. Then in relation (5.12) we have that Γ+ (A1 )F −1 ϕ ∈ H ⇐⇒ Γ+ (A)F −1 ψ ∈ H. If condition (5.14) where ψ is given by the expression (5.15) holds then relation (5.12) holds too. Therefore F −1 ϕ ∈ D(W+ (B, A1 )), F −1 ψ ∈ D(W+ (B, A)) and, consequently, ϕ ∈ D(X+ (A, A1 )). Due to Corollary 3.8 we have FD(W+ (B, A1 )) ∩ Ha,b = Ha,b .

Therefore, as a result, the estimate

codimD(X+ (A, A1 )) < ∞

is proved. In addition, the operator X+ (A, A1 ) : H → H is bounded. If we repeat the proof for the pair (A1 , A) instead (A, A1 ) then we obtain that codimR(X+ (A, A1 )) < ∞

and the inverse operator X+ (A, A1 )−1 : H → H is bounded too. Consequently D(X+ (A, A1 ))D(X+ (A, A1 )),

R(X+ (A, A1 ))R(X+ (A, A1 ))

what signifies that the subspaces M+ = D(W+ (A, A1 )) and N+ = R(W+ (A, A1 )) are closed and have finite codimension. The theorem is proved. Lemma 5.5. The following estimate holds (see (5.4), (5.18)) ∞ ∞ ρ(τ ) dsdτ < ∞. |RM (s, τ )|2 ρ(s) 0

(5.25)

0

The proof is based on the properties of the operator Rτ in the class of the considered functions (see, e.g., (4.9)) From Definition 5.2 and Theorem 5.4 results the following Corollary. Corollary 5.6. If z ∈ D(W+ (B, A1 )) then the equality

W+ (B, A)w = W+ (B, A1 )z

holds iff w = W+ (A, A1 )z, where z ∈ D(W+ (A, A1 )).

(5.26)

Wave Operators for Nonlocal Sturm-Liouville Operators Proposition 5.7. Let s ∈ (−∞, ∞). Then the following inclusions eisA Hess (A) ⊂ Hess (A)

eisA D(W+ (B, A)) ⊂ D(W+ (B, A))

and the relation hold.

e−isB W+ (B, A)zW+ (B, A)e−isA z, z ∈ D(W+ (B, A))

63

(5.27) (5.28) (5.29)

Proof. It is known that eisA Rζ (A) = Rζ (A)eisA , ζ ∈ ρ(A).

isA

The operator e is bounded. Consequently inclusion (5.27) results from Definition 3.1. Consider now the identity ei(t+s)B e−i(t+s)A z = eisB (eitB e−itA )e−isA z. Suppose z ∈ D(A) ∩ Hess (A). As eisA D(A) ⊂ D(A) then in view of (5.27) e−isA z ∈ D(A) ∩ Hess (A).

(5.30) (5.31)

If additionally Γ± z ∈ H0 ∩ H then z ∈ D(W+0 (B, A)) and in the left side of (5.30) there exists the limit if t → +∞ (see Prop. 3.4). The existence of the limit in the right side of (5.35) and condition (5.31) give (using once more Proposition 3.4) Γ+ (e−isA z) ∈ H0 ∩ H. Consequently, e−isA z ∈ D(W+0 (B, A)) and e−isB W+0 (B, A)z = W+0 (B, A)e−isA z.

The operators e

−isB

, e−isA are bounded, so as corollary, the relation is proved.

Theorem 5.8. If z ∈ D(W+ (A,A1 )) ∩ D(W+ (B,A1 )) then eisA1 z ∈ D(W+ (A,A1 )) ∩ D(W+ (B,A)) and eisA W+ (A, A1 )z = W+ (A, A1 )eisA1 z.

(5.32)

Proof. Proposition 5.7 for the operators A and A1 respectively gives and

eisB W+ (B, A)w = W+ (B, A)eisA w, w ∈ D(W+ (B, A)) eisB W+ (B, A1 )z = W+ (B, A1 )eisA1 z, z ∈ D(W+ (B, A1 )). Suppose z ∈ D(W+ (A, A1 )) ∩ D(W+ (B, A1 )). Then w = W+ (A, A1 )z ∈ D(W+ (B, A))

(see Corollary 5.6). Substituting into (5.33) we obtain

eisB W+ (B, A)W+ (A, A1 )z = W+ (B, A)eisA W+ (A, A1 )z. The use of relation (5.26) in the left side gives eisB W+ (B, A1 )z = W+ (B, A)eisA W+ (A, A1 )z. The comparison with (5.34) gives W+ (B, A)eisA W+ (A, A1 )z = W+ (B, A1 )eisA1 z.

(5.33) (5.34)

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E.V. Cheremnikh

Due to Corollary 5.6 we obtain that eisA1 z ∈ D(W+ (A, A1 )) ∩ D(W+ (B, A1 )) and that relation (5.32) holds. The theorem is proved. We will discuss the subspaces (see Theorem 5.4) Hess (A1 |A) = D(W+ (A, A1 )), Hess (A|A1 ) = R(W+ (A, A1 )). For this aim we will consider a family of operators A with one spectral singularity σ0 ∈ (0, ∞) of multiplicity 1. Then δ− (σ0 )δ+ (σ0 ) = 0 and from equation (5.17) it follows (if τ = σ0 ) ∞ M (s, σ0 )θ(s)ds = f (σ0 ). (5.35) 0

Using the decompositions of the functions M (s, τ ), f (τ ) according to the general formula (see (5.4)) h(τ ) = h(σ0 )

σ0 + 1 τ − σ0 + Rh(τ ), Rh(τ ) = h(τ ) + (σ0 + 1)Rσ0 h(τ ) τ +1 τ +1

we obtain (see (5.21)) ξ(τ )θ(τ ) +

∞ 0

RM (s, τ )θ(s)ds = Rf (τ ), ξ(τ ) =

τ +1 δ− (τ )δ+ (τ ). τ − σ0

The solution θ(τ ) defines a bounded operator Φ : L2ρ (0, ∞) → L2ρ (0, ∞), i.e., θ = Φ(Rf ). Substituting θ into (5.35) we obtain ⎤ ⎡ $ # ∞ ∗ ∞ ϕ (s) f (s) ∗ ds = f (σ0 ) ⎣1 + (σ0 + 1)V.p. ds⎦ V.p. ϕ (s) f (s) + (σ0 + 1) s − σ0 s − σ0 0

0

(5.36) where ϕ (s) = Φ (M (·, σ0 ))(s). In the concrete case we consider the operator A (see (2.2)) generated by an element η ∈ L2 (0, ∞) such that the transformation γ = Fη is ∗

∗

γ(τ ) =

−k , k > 0, σ0 > 0. √ τ − ( σ0 + ik)2

(5.37)

By analogy, an other value k1 defines the function γ1 (τ ) and the operator A1 . Such operators have a unique spectral singularity σ0 which is not an eigenvalue (because the equation −y ′′ = σ0 y has no solution in L2 (0, ∞)). Obviously √ η(x) = −kei( σ0 +ik)x (compare (3.18)), so, the element η does not satisfy conditions (2.4)–(2.5). But we need (2.4)–(2.5) in the general case of η only to obtain


65

the differentiability of some auxiliary function. In the concrete case (5.37) we can repeat again the statements of Section 2. Note that γ(s)ρ(s) − γ(τ )ρ(τ ) s−τ # 1 1 k √ √ √ =− √ √ √ 2π s + τ ( s − σ0 − ik)( τ − σ0 − ik) $ 1 √ . + √ √ √ ( s + σ0 + ik)( τ + σ0 + ik)

Rτ (γρ)(s) =

(5.38)

A simple calculus gives √ √ √ √ τ − σ0 τ + σ0 δ+ (τ ) = √ , δ+ (τ ) = √ √ √ τ − σ0 + ik τ + σ0 − ik and b(τ ) =

√ 1 −k τ . (δ+ (τ ) − δ− (τ )) = √ 2i τ − σ0 + 2ik σ0 + k 2

(5.39)

√ √ √ √ τ − σ0 + ik1 τ + σ0 − ik S1 (τ ) √ =√ √ √ S(τ ) τ + σ0 − ik1 τ − σ0 + ik

(5.40)

If S(τ ) = δ− (τ )/δ+ (τ ) then I(τ ) = and

# √ √ √ √ Rτ (I)(s) = 2i(k − k1 ) ( s + τ )( s + σ0 − ik1 )

$−1 √ √ √ √ √ √ ×( τ + σ0 − ik1 )( s − σ0 + ik)( τ − σ0 + ik) .

Let

F± ϕ(τ ) =

(5.41)

1 1 ϕ(τ ) ± H[ϕ](τ ). 2 2i

Lemma 5.9. The expression f (τ ) (see (5.19)) takes the form √ ∞ ∞ √ τ − σ0 − ik1 1 Rτ (γ1 ρ)(s)ϕ(s)ds + b(τ )(I(τ ) − 1) Rτ lnsϕ(s)ds f (τ ) = − √ √ τ − σ0 + ik 2i 0

+

∞# 0

0

$ 1 Rτ (γρ)(s)F− ϕ(s) + Rτ (γρ)(s)I(s) + b(τ )Rτ (I)(s) F+ ϕ(s) ds. π (5.42)

The proof reduces to algebraic transformations and the relation ∞ 0

ϕ(τ )H[ψ](τ )dτ = −

∞ 0

H[ϕ](τ )ψ(τ )dτ.

66

E.V. Cheremnikh Obviously, if τ = σ0 then

f (σ0 ) = − +

∞ # 0

k1 k

∞ 0

Rσ0 (γ1 ρ)(s)ϕ(s)ds +

1 b(σ0 )(I(σ0 ) − 1) 2i

∞ 0

Rσ0 ln sϕ(s)ds

$ 1 Rσ0 (γρ)(s)F− ϕ(s) + Rσ0 (γρ)(s)I(s) + b(σ0 )Rσ0 (I)(s) F+ ϕ(s) ds. π (5.43)

Further we suppose that √ ik1 (ε) = −2 σ0 + iε,

ε > 0,

ε → 0.

(5.44)

Then in view of (5.40) I(σ0 ) =

M1 (ε) , ε

M1 (0) =

√ 2 σ0 √ (2 σ0 − ik) = 0 k

and in view of (5.41) Rσ0 (I)(s)

Q(s) M2 (ε) √ , √ ε s − σ0 − iε

√ M2 (0) = −4i σ0 − 2k = 0

where 1 √ . Q(s) = √ √ √ ( s + σ0 )( s − σ0 + ik) Now we fix the value k such big that the operator Φ exists and equation (5.36) holds. In this equation the expression f (τ ) only depends on k1 . Taking into account this dependence, condition (5.44) and the relations (5.38)–(5.41) we obtain for the expression (5.43) the following decomposition 1 f (σ0 ) = f1 (σ0 , ε) + f2 (σ0 , ε), ε

f1,2 (σ0 , ε) = O(1), ε → +0.

(5.45)

Note that # ∞ 1 f2 (σ0 , ε) = b(σ0 ) M1 (ε) Rσ0 ln sϕ(s)ds 2i 0

1 + M2 (ε) π

∞ 0

√

$ Q(s) F+ ϕ(s)ds . √ s − σ0 − iε

Therefore we have a subset N dense in L2ρ (0, ∞) (maybe changing k) such that f2 (σ0 , 0) = 0, ϕ ∈ N .


67

Theorem 5.10. There exist the operators A, A1 , A2 of type (2.2) with the same set of spectral singularities, S(A) = S(A1 ) = S(A2 ), such that Hess (A|A1 ) = Hess (A|A2 )

(5.46)

Proof. It is sufficient to consider the examples stated above with the same unique spectral singularity σ0 . Let the values k = k1 be sufficiently big, A, A1 be corresponding operators. Let ϕ be a fixed element such that ϕ = 0, ϕ ∈ Hess (A|A1 ). Then equation (5.36) holds. Suppose that k1 varies according to (5.44), let k2 = k1 (ε). If the relation Hess (A|A1 ) = Hess (A|A2 ) holds always then equation (5.36) holds too. But this is impossible. Really, due to the form (5.37) the function ϕ∗ (s) is differentiable, in view of (5.42) and also (5.38)–(5.41) the left side of the relation (5.36) is bounded if ε → +0. The right side in (5.36) is unbounded (see (5.45)) for ϕ ∈ N . Obviously for small ε the relation (5.41) is disturbed too for the element ϕ ) ∈ N ∩B, where B = {ϕ ) : ||ϕ ) − ϕ|| < r} is some ball. So, N ∩B ∩Hess (A|A2 ) = ∅. As the space Hess (A|A2 ) is closed then N ∩ B∩Hess (A|A2 ) = ∅ or B∩Hess (A|A2 ) = ∅ what proves (5.46). The theorem is proved.

Conclusion In this article we consider the wave operator W± (A2 , A1 ) = s − lim eitA2 e−itA1 . t→±∞

Suppose that the operators A1 , A2 are selfadjoint, and the wave operators W+ (A1 , A2 ) and W+ (A2 , A1 ) exist. Then the wave operator W+ (A2 , A1 ) : H1,ac → H2,ac is isometric and transforms the whole of H1,ac onto the whole of H2,ac (see [12], Chap. 10, 12.3). We study non-selfadjoint case, namely Sturm-Liouville operators with non local boundary conditions. For such operators 1) the wave operator W+ (A1 , A1 ) may be defined as bounded operator if the sets of spectral singularities of A1 and A2 coincide (Theorem 5.4) and 2) the domain of definition D(W+ (A1 , A2 )) in the general case varies simultaneously with the variation of the operator A2 (Theorem 5.10). Therefore in the non-selfadjoint case it is expediently to define a so-called relative essential subspace Hess (A1 |A2 ) instead to define some subspace Hess (A1 ) for one isolated operator A1 . The idea itself to use the Laplace transform was taken from work [10]. Practically we use the Friedrichs’ model. One can find the operators with non-local condition within the Friedrichs’ model in [9] too.

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References [1] Birman M.Sh. On perturbation of the spectrum of singular operator under the variation of the boundary condition // Dokl. Akad. SSSR, 1961, n. 137, 761–763 (in Russian). [2] Sachnovich L.A. Dissipative operators with absolutely continuous spectrum, Trudy Moscow Math. Soc., 1968, v. 19, 211–269 (in Russian). [3] Pavlov B.S. On the conditions of separability of spectral components for dissipative operators // Izv. Akad. SSSR, ser. math., 1975, v. 39, n.1, 123–148 (in Russian). [4] Naboko S.N. Functional model of perturbation theory and its applications to the scattering theory // Trudy Moscow Math. Soc., 1980, n. 147, 86–114 (in Russian). [5] Pavlov B.S. Spectral analysis of differential operators with “spreaded” boundary condition // Probl. Math. Phys., 1973, n. 6, 101–119 (in Russian). [6] Kato T. Wave operators and similarity for some non-selfadjoint operators // Math. Ann., 1966, n. 162, 258–279. [7] Kako T., Yajima K. Spectral and scattering theory for a class of non-selfadjoint operators // Sci. Par. Coll. Gen. Educ. Univ. Tokyo, 1976, v. 26, n. 2, 73–89. [8] Cheremnikh E.V. Nonselfadjoint Friedrichs’ model and Weyl function // Dokl. Akad. Ukraine, 2001, n. 8, 22–29 (in Ukrainian). [9] Cheremnikh E.V. Friedrichs’ model and the problems with non-local boundary conditions // Math. Meth. and Phys.–Mech. Fields, 2000, v. 43, n. 3, 146–156 (in Ukrainian). [10] Alsholm Preben Inverse scattering theory for perturbation rank one // Duke Math. J., 1980, v. 47, n. 2, 11–23. [11] Diaba F., Cheremnikh E. On asymptotic time for an evolution with non-local boundary condition // J. Dynam. Syst. and Geom. Theories, 2006, v. 4, n. 1, 61–77. [12] Kato T. Perturbation theory for linear operators, Springer-Verlag, 1966. [13] Yosida K. Functional analysis, Springer-Verlag, 1965. E.V. Cheremnikh Gnatjuka Str. 8, Flat 12A Lviv 79007, Ukraine e-mail: [email protected]


Semiclassical Results for Ideal Fermion Systems. A Review Monique Combescure Abstract. Ideal Fermion systems represent an issue of great recent interest. They consist of N-Fermion gases without interaction but subject to an exterior field or potential. In this paper we review recent studies of such systems in the following situations: • the low temprerature behavior of confined fermion gases in contact with an exterior reservoir • the action of a magnetic field on two-dimensional electronic devices • the transport and dissipation for time-dependent scatterers connected to several leads, known as “quantum pumps”. The response of these different systems under the influence of the exterior potential or field is considered. In the first situation the response is calculated in terms of the “dynamical susceptibility” in the linear response framework, which yields a generalized Kubo formula. In the second situation the response is expressed by the magnetic susceptibility and the magnetization which are estimated semiclassically for various regimes of the temperature as compared with powers of the Planck constant. In the last situation the response is a measurable current estimated in an adiabatic framework. All these results are contained by a series of papers by D. Robert and myself, and in recent works by Avron, Elgart, Graf and Sadun. Mathematics Subject Classification (2000). 81-02, 81Q20, 81R30, 82B10. Keywords. Semiclassical analysis, coherent states methods, fermionic systems, linear response theory.

1. Introduction The study of ideal fermion gases and of their response in several physical situations has focused a large interest in the recent years. The various situations involved are for example: (i) external time-dependent perturbations of confined fermion gases in contact with an external reservoir;

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M. Combescure

(ii) the action of a magnetic field on two-dimensional electronic devices; (iii) the transport and dissipation for so-called “quantum pumps” modelled by time-dependent scatterers connected to several leads. The reservoirs (or leads) with which these systems are in contact are at some non-zero temperature T . In the case (i), the “response” is measured via the socalled “dynamical susceptibility” and yields a Generalized Kubo Formula. In case (ii), the response is the “magnetic susceptibility” which presents interesting and measurable features in the so-called “mesoscopic devices”. In case (iii) the response is a measurable current. These topics have focused large attention in the physical literature, (see the bibliography), and led to understand the underlying physical phenomena. However until recently, only a few rigorous studies have been attempted so far. (Note [29] where a rigorous semiclassical study of thermodynamic properties of Fermion or Boson gases is performed when the Fermions are confined via a power-law potential; see also [22].) Recently, some mathematical work has been devoted to the three situations considered above: • The so-called Linear Response Theory; • The semiclassical regime where some small parameter (usually the Planck constant) is assumed to tend to zero; • The adiabatic regime (for case (iii)), where adiabaticity of the external timedependent perturbation is considered small (and thus can be treated as a “semiclassical parameter”). In this review, we shall present mathematical results for the three cases (i)– (iii). They will be based respectively on [10], [9], [2]. Of course this review is not exhaustive in the sense that many other interesting physical aspects have been considered but have not, to our knowledge, received so far an attention towards their rigorous treatment. Let us mention a few of them, with bibliographic references, so that interested mathematical-physicists might find a source of inspiration: • The dielectric response of dilute electron gases in small semiconductor devices ([33], [27]); • Static semiclassical response of a bounded electron gas ([20], [26]); • The dynamics of ideal Fermion gases in the presence of a combined timedependent harmonic potential, and a periodic potential ([6]); • The Friedel oscillations at zero temperature of the particle density-distribution for ideal fermion gases in one-dimensional harmonic traps ([15]); • Quasi one-dimensional electron gas in a semiconductor quantum wire in the presence of some random field ([27]); • A kinetic semiclassical equation for modelling transport in degenerate gases ([24], [23]);

Semiclassical Results for Ideal Fermion Systems

71

• Ground-state number fluctuations of trapped ideal and interacting fermion gases ([31], [32]); • the “Riemannium”, i.e., a chaotic Fermion gas whose single-particle energylevels are given by the (imaginary part of) the complex zeros of the Riemann zeta function ([19]); • The eigenvalue statistics of quantum ideal gases with single-particle energylevels of the form En = nα ([11]); • The behavior of a degenerate quantum system as a precursor to a “paired Fermi condensate” analogous the the Bose-Einsten Condensate, at very small temperature ([8]); • The link of Fermi-Dirac statistics with the number theory ([17]).

The plan of this review article is as follows:

In Section 2 we develop a rigorous semiclassical expansion of the “dynamical susceptibility” for an ideal confined fermion gas, at small temperature, assuming some chaoticity assumption of the one-particle motion on the Fermi energy surface [10]. In Section 3, we consider an electron gas in any dimension, subject to an inhomogeneous magnetic field, in a semiclassical regime, and examine various temperature regimes where the magnetic response can be evaluated. This allows a description of the “de Haas-van Alphen” oscillations [9]. In Section 4 we give the results on Transport and Dissipation in “Quantum Pumps” as described by the work of Avron-Elgart-Graf-Sadun [2].

2. Semiclassical results in linear response theory Let us consider a system of non-interacting fermions • confined by potential, • in contact with exterior reservoir at temperature T .

We assume that a time-varying potential drives the system – out of EQUILIBRIUM, – but close to it. The Response to this external perturbation is measured by the dynamical susceptibility. The complete rigorous analysis is difficult in that it involves the non-equilibrium statistical mechanics. Here we shall follow a semi-empirical route (small perturbations) and we try to solve this problem “to first order in perturbation”, namely in the Linear Response Framework. – We assume some “chaoticity assumption” on the one-body classical dynamics; – We derive a generalized Kubo Formula; – We perform semiclassical expansions in when → 0;

72

M. Combescure – We exhibit a low temperature limit, where the closed classical trajectories manifest themselves in the Response function (or in the Dynamical Susceptibility).

The model is as follows: we start with a classical Hamiltonian of the form: p2 + V (q) in Rn , 2m where the potential is confining and smooth enough: H(q, p) =

V (q) ≥ c0 (1 + q 2 )s/2 ,

c0 , s > 0,

α

|∂ V (q)| ≤ Cα V (q), Cα > 0, ∀α ∈ Nn , ˆ H is the Weyl quantization of H; it is a self-adjoint operator on H := L2 (Rn ) The assumptions imply that sp(H) ⊂ [0, ∞) is pure-point: ˆ j = Ej ψj , j ∈ N, ψj ∈ H. Hψ

The infinite system of fermions is described via states in the Fock Space: F := n n ⊕∞ n=0 ⊗a H, where ⊗a H antisymetric tensor product of H. It represents the space of n fermionic states. The framework is thus the second quantization: *ˆ ˆ acting on F H := dΓ(H)

second quantized Hamiltonian,

*ˆ N := dΓ(1lH ) number of particles operator,

nj ∈ {0, 1} is the occupation number of any state ψj in F (which expresses the Pauli principle); *l,n (ψj1 ∧ · · · ∧ ψjn ) = nl (ψj1 ∧ · · · ∧ ψjn ), ∀l = j1 , . . . , jn , N

tells whether or not the level El is occupied in a given state of ⊗na H. Define: *j := ⊕∞ N *j,n , N n=0

** N =

∞ j=0

*j , N

** H =

∞ j=0

ˆj Ej , N

$ # ** ** N = 0. H,

The Grand-Canonical Partition function is:

*ˆ *ˆ ZG := Tr eβ H+κN ,

1 , κβµ, T is the temperature, and µ is the chemical potential. where β = kT The trace is taken in F. Thus ZG can be rewritten as: ∞ + ,−1 1 + e−β(Ej −μ) . ZG = j=0


73

Let f be the Fermi-Dirac function f (x) := (1 + ex )−1 . The operator in H: . ˆ − µ) ρêq := f β(H

is called the Fermi-Dirac one-body operator. Now assume that the one-body Hamiltonian is slightly perturbed in a timedependent way: ˆ λ (t) := H ˆ + λAF ˆ (t) H ˆ λ is a small parameter, A is some self-adjoint operator, and F is of the form, with α > 0: / t 0

(confinement assumption). * κ be the Weyl quantization of Hκ . The previous assumptions ensure Let now H *κ ) ⊂ [ε, ∞) is pure point for every * κ is self-adjoint and its spectrum σ(H that H κ ∈ R where ε > 0. Let us call (Ej )j∈N and (ϕj )j∈N the set of corresponding eigenvalues and eigenstates. Given β > 0, we set: 1 (3.3) Fβ (x) = − Log 1 + e−βx , β −1 fβ (x) = Fβ′ (x) = 1 + eβx , (3.4) fβ is related with the Fermi-Dirac distribution. These functions are meromorphic, with poles (or cuts for Fβ ) at: x=

2j + 1 iπ, β

j ∈ Z.

In the grand-canonical ensemble, the thermodynamic potential Ω is given by: . *κ − µ , Ω(β, µ, κ) = (3.5) Fβ (Ej − µ) = tr Fβ H j∈N

where µ > 0 is the chemical potential and β is given by (3.1). Furthermore, the mean-number of particles in the grand-canonical ensemble is given by: .. - *κ − µ . (3.6) N (β, µ, κ) = tr fβ H

In all that follows, we fix the chemical potential µ, which according to (3.6) implies that N (β, µ, κ) is large in the semiclassical régime so that we are approaching the thermodynamical limit where canonical and grand canonical ensemble descriptions * −µ) are equivalent. Using the functional calculus, it is not difficult to see that Fβ (H * − µ) are trace-class and that the function : κ → Ω(β, µ, κ) is C ∞ for and fβ (H |κ| ≤ κ0 . ∂ . We shall denote ∂κ = ∂κ


79

Proposition 3.1. The function: κ → Ω(β, µ, κ) is C ∞ on R. In particular we have , . + * κ − µ ∂κ H *κ . (3.7) ∂κ Ω = tr fβ H

Now we have the following definitions of magnetization M and magnetic susceptibility χ: , . + * κ − µ ∂κ H *κ , M = ∂κ Ω = tr fβ H (3.8) χ =

∂κ M.

(3.9)

We shall now distinguish different Temperature Regimes, where the magnetic response behaves semiclassically in significantly different ways: • When T is larger than we have asymptotic expansions in of the Thermodynamic Grand-Canonical Function, and thus of the Magnetic Response functions. As a Corollary, we recover the Landau diamagnetic formula for 2-dimensional ideal electron gases. • When T is of the order of , we prove that the magnetization and the magnetic susceptibility split into two parts: an average part with a regular asymptotic expansion in , plus an oscillating part in which represents the contribution of the classical periodic orbits that lie on the energy surface at the Fermi level µ. The first Temperature Regime we consider is β ≤ −2/3+ε for some ε > 0: Theorem 3.2. For any ε > 0 and κ0 > 0, Ω admits an asymptotic expansion in , uniform in κ for |κ| ≤ κ0 , and for β ≤ ε−2/3 . More explicitly, for any N ∈ N we have: N . 3N (−1)k+1 j Ω = h−n Ωjk + O N +1−n β 2 +k(n) , (3.10) k! 3j j=0 k≤

2

with Ωjk =

(k)

IR2n

dq dp djk (q, p) Fβ (Hκ − µ),

djk being a suitable linear combination of derivatives of Hκ with respect to q, p and k(n) a constant depending only on the dimension n (k(n) ≤ 2n + 1). In particular: Ω00 = dq dp Fβ (Hκ (q, p) − µ) , (3.11) IR2n

1 1 β Ω22 − Ω23 = − 2 6 48π 2 where Bjk is the magnetic field 2 B 2 = Bjk , j 0 is rather technical. We can extend the above results to a more natural temperature Regime β ≤ −1+ε : Assume in addition to (H1–H3): (H4) µ is non-critical for Hκ . Then we have: Theorem 3.4. Assume (H1-4). Then the magnetization M = ∂κ Ω has for any 2 temperature T satisfying 1−ε ≤ T ≤ 3 −ε (some ε > 0) a complete asymptotic expansion in obtained by taking the derivative in κ of the formal expansion in for Ω given (3.10). We now come to the so-called “mesoscopic regime” where T ∼ . We introduce a new parameter σ: σ := β, (3.15) as in the previous section. Let Σκμ be the energy surface at energy µ = Fermi-level (or equivalently the chemical potential): Σκμ := (q, p) : in R2n , Hκ (q, p) = µ ,

and let dσμκ be the Liouville measure on Σκμ .

Then we assume the following (Gutzwiller Assumption): (H5):

on Σκμ the closed orbits of period smaller than some prescribed τ > 0 are nondegenerate.

We shall define a “regularized” magnetization as follows: Let ρ ∈ C0∞ be an even function such that: ρ(t) ≡ 1,

if

|t| ≤ 1;

ρ(t) ≡ 0,

if |t| ≥ 2 and

Let us define for τ > 0, ρτ (t) = ρ(t/τ ),

ρ(t)dt = 1.

(3.16)

R

(3.17)

Semiclassical Results for Ideal Fermion Systems and

/

Mτ = Tr (fσ ∗ ρ)τ )

0

*κ − µ H

1

*κ ∂κ H

2

,

81

(3.18)

where ) g denotes the inverse Fourier transform of g. Clearly Mτ → M when τ → ∞. However we shall not be able to let τ → ∞ in this work. This is the same type of restriction we had in the previous Section, where only a “regularized dynamical susceptibility” has been estimated. Then our result is as follows: Theorem 3.5. Let us assume H-1 to H-5 and σ = β ∈ [σ0 , σ1 ] where σ1 > σ0 > 0 are fixed. For χτ = χ ∗ ρτ , τ > 0, we have the decomposition with χ=−

2−n 24π 2

χosc =

κ

μ

e

γ∈(Γμ )τ ∞

(3.19)

(3.20)

dσμκ B(q) 2 +

i(Sγ /+νγ

+ O( ),

χτ = χ + χosc ,

π 2)

cχ,k (µ, κ, σ, T )k + O(∞ ),

k≥3−n

⎧ ⎨

⎫ ⎬ rγ m2γ /2σ ρ1,τ (Tγ ) (k) k + dχ,γ 1/2 ⎩ | det(1 − Pγ )| ⎭ sinh(πTγ /σ) k≥1

(3.21)

where we use the following notations: • γ is a periodic orbit in classical phase-space, Sγ the classical action along this orbit, • Tγ is the period of the orbit γ, • rγ ∈ Z the repetition number of this orbit, and • Pγ the linearized Poincaré map of the orbit γ (which doesn’t have the eigenvalue 1 because of (H5)). Furthermore Tγ∗ dt ∂κ Hκ (qt , pt ), where Tγ ∗ := Tγ /rγ is the period of the primitive • mγ := 0

orbit, • mγ represents the flux of the magnetic field through the closed curve γ, • cχ,γ , dχ,γ are smooth coefficients depending on the periodic orbit γ, on σ, and on the function ρ. The proof is rather involved (see [9]).

To complete this section let us show that a similar decomposition of the ¯ and a highly oscillating part Mosc magnetization M into a “smooth” part M holds true, and that indeed the dominant term in the semiclassical expansion of ¯ is nothing but the well-known diamagnetic Landau term. M

82

M. Combescure

Proposition 3.6. Assume (H1-4). Then for any σ1 > 0 and for κ0 > 0, uniformly for β = σ ∈]0, σ1 [ we have, mod O(h∞ ), 2−n M = −κ dσ 0 B(q) 2 + ck (µ, 0, σ, T )k + O(∞ ), (3.22) 24π 2 0μ μ k≥3−n

where field.

dσμ0

is the Liouville measure on the energy surface Σ0μ , i.e., without magnetic

Conclusion The result of Theorem 3.5 is very similar to the one obtained in Theorem 2.2 for the dynamical susceptibility in the Linear Response Regime for an ideal Fermion gas. The temperature dependence occurs via the same parameter σ = β, and the contribution of closed orbits on the Fermi Energy Surface is strongly damped by the factor 1 . sinh(πTγ /σ) Natural classical quantities appear in the dominant parts of the expansions, namely here, instead of the Classical Correlation Functions on Energy Surface or along the closed orbits, we have:

• The diamagnetic Landau term Σ0 dσμ0 B(q)2 , μ • The Flux mγ of the magnetic field through the closed orbits, the periods of the closed orbits.

4. Transport and dissipations in quantum pumps This section is devoted to a presentation of the work of [2], (see also [3]). We think that it is pertinent in this review paper, since it presents interesting scattering features for the electron gas under consideration, whereas the two previous sections dealt with ideal confined fermion gases where the spectrum of the single-particle Hamiltonian was thus pure point. The Hamiltonian of a single-particle is here of spectrally continuous nature. Moreover it is driven via a time-dependent potential, which is assumed adiabatic. That means that there is an adiabatic time-scale called ε−1 which is considered large with respect to the dwell time τ of the particles into the scatterer. Thus the parameter ε plays the rˆ ole of a “semiclassical parameter”, although here is taken equal to 1. An adiabatic quantum pump is a slowly time-varying scatterer connected to several leads. Each lead may have several channels. The total number of channels in each lead will be denoted by n. Each channel is represented by a semi-infinite, onedimensional, single mode ideal wire. It is assumed that the particles propagating in each channel are non-interacting and have all energy k 2 /2; along each lead, electrons can enter or leave the pump.


83

Then the question is: How electrons are transported from one lead to another due to the timedependent potential of the scatterer (hence the name of “pump”)? This question was first addressed in [7] in the framework of Linear Response Theory. They established that the expected current is a function of the scattering matrix and of its time-derivative. The interest in the subject was piqued by the experiments ([30], although it is not completely clear that the experiment is actually represented by an adiabatic pump). The mathematical framework is as follows: • The single-particle Hilbert space is given by

H = H0 ⊕ L2 (R+ , Cn ),

where states in L (R+ , C ) := ⊕nj=1 L2 (R+ ) (resp. H0 ) describe electrons in the leads (resp. in the pump). Denote by Πj , j = 0, 1, . . . , n, H → H the projection onto H0 if j = 0, and onto the jth copy of L2 (R+ ) if j = 1, . . . , n. • The Hamiltonian is a slowly varying time-dependent self-adjoint operator in H, ˆ ε (s) ≡ H(ε ˆ −1 s) where ε is the small adiabatic parameter. The namely given by H ˆ assumptions on H(t) will ensure the existence of a unitary propagator in H, called odinger equation: Uε (s, s′ ) satisfying the Schr¨ 2

n

ˆ ε (s)Uε (s, s′ ). i∂s Uε (s, s′ ) = ε−1 H ˆ ǫ (s) are as follows: • The assumptions on H ˆ ˆ (H1): Hε (s) − Hε (s′ ) is bounded in H and smooth in s, s′ , ˆ ε (s) + i)−m 1 < C for all s, ε, and some m ∈ N, (H2): Π0 (H ˆ ε (s)ψ = −d2 ψ/dx2 for ψ ∈ C ∞ (R+ , Cn ), (H3) H 0 ˆ ε (s)) ∩ (0, ∞) is empty, (H4) σpp (H ˆ ε (s) ≡ H ˆ − , ∀s ≤ 0. (H5) H

(Here · 1 denotes the trace class norm on H.) • The electrons in the incoming state should be in an equilibrium state. We take ˆ − ), where ρ is a function of bounded variation with it as a density matrix ρˆ := ρ(H Supp(ρ) ⊂ (0, ∞). A good example for it is the zero-temperature equilibrium state at Fermi-level µ: ρ(λ) := Θ(µ − λ), where Θ is the Heaviside function. • Units are chosen so that k = = m = e = 1 where k is the Boltzmann constant, and m, e the mass and charge of the electron. • One can now define the “charge current” operator. Consider the following operator in H: d 1 d A := 0 ⊕ ( v(x) + v(x) ), 2i dx dx where v : [0, ∞) → R is smooth, with v ′ (x) ≥ 0, Supp(v) ⊂ [b, ∞), some b > 0, and v(x) = x for x > c > b. A is self-adjoint and commutes with Πj , ∀j = 0, 1, . . . , n, and we denote Aj := AΠj .

84

M. Combescure

The operator A allows to distinguish incoming and outgoing electrons. Take some a > 0 and a “switch function” f ∈ C0∞ where: f (x) = 0, for x < −1, f (x) = 1, for x > 1. The charge current operator is then defined as ˆ ε (s) , f (Aj − a) + f (−Aj − a)] ≡ Ij+ (a) + Ij− (a). Ij (a) := i[H

• The expectation value of the current at “epoch” s in the channel j, is then given by ˆ − )Uε (−1, s)Ij (a)). I j (s, aε) := tr(Uε (s, −1)ρ(H ˆ0 • In order to define “scattering” by the pump, some reference Hamiltonian H has to be introduced. Take 2 2 ˆ 0 := h ⊕ (− d )N ⊕ · · · ⊕ (− d )N , H 2 dx dx2 ˆ 0 on each of the leads is (− d22 )N which is simply the where the component of H dx Laplacian with Neumann boundary conditions. h for the pump component can be ˆ anything, provided that H0 obeys (H2). The time-dependent S-matrix is given by S(s) :=

lim

s± →±∞

U0 (ε−1 (s − s+ ))Uε (s+ , s− )U0 (ε−1 (s− − s)).

(4.23)

It is the so-called frozen scattering operator usually introduced in adiabatic scatˆ 0 ) is conserved so that tering theory. “Energy” E (the spectral representation for H the E-fibers S(s, E) of the operator S(s) are j n × n “scattering matrices”. Then everything is in order to state the result: Theorem 4.1. Consider a cut-off function in energy space χ ∈ C0∞ (0, ∞) with χ = 1 on Supp(ρ′ ). Redefine the current operator (with IR and UV cutoff): ˆ ε (s))i[H ˆ ε (s), f (±Aj − a)]χ(H ˆ ε (s)). I,,± (s, a, ε) := χ(H

Then, under the above assumptions, we have the following expression for the current in the jth channel:

∞ dS i lim lim ε−1 I j (s, a, ε) = − , (4.24) dEρ′ (E) S(s) a→∞ ε→0 2π 0 ds j,j where S(s) ≡ S(E, s) (the E-fiber of the scattering operator S(s)). The double limit is uniform in s ∈ I, I being a compact interval. The proof is given in [2]. Remark 4.2. The physical meaning of the limiting procedure of (4.24) is that the current measurement is made well outside the scattering region (lima→∞ ), but after the adiabatic limit (limε→0 ). Further results of ref. [3] are: • The relationship with classical scattering, • The formulation of charge transport in terms of the Chern character of a natural line bundle.


85

Conclusion Semiclassical tools appear useful in the context of an adiabatically driven electron system. Under appropriate mathematical assumptions, it provides the expected physical result: namely a time-dependent scatterer connected to several leads acts as a “quantum pump” that allows charge transfer between the leads. This is the response of the scattering system to a slowly varying potential. The condition that Supp(ρ) ⊂ (0, ∞) might seem strange, since it excludes the natural Fermi-Dirac density matrix at positive temperature T : .−1 ˆ ˆ := 1 + eβ(H−μ) ρ(H) , (4.25)

where as in previous sections µ is the chemical potential, and T ≡ β −1 is the temperature (although this restriction is not present in [3]). It seems rather technical, and the link with physics (exterior leads at any temperature T ) should be clearer if it is removed. We think that the introduction of cutoff function in energy in Theorem 4.1 should allow to remove this condition. Thus this system provides a simple example where the scattering of an electron system leads to a measurable response (whereas studies of previous sections were restricted to bound state systems). If the pump were a “chaotic billiard” the classical dynamics will in general have arbitrarily long orbits, which should give rise to resonances, although this point has not been studied yet mathematically, to our knowledge.

References [1] Agam O., J Phys. I (France), 4, 697–730, (1994). [2] Avron J., Elgart A., Graf G.M., Sadun L. Schnee K., Adiabatic charge pumping in open quantum systems, Comm. Pre Appl. Math., 57, 528, (2004). [3] Avron J., Elgart A., Graf G.M., Sadun L., Transport and Dissipation in “Quantum Pumps”, J. Stat. Phys. 116, 425–473, (2004). [4] Avron J., Gutkin B., Oaknin D., Adiabatic Swimming in an Ideal Quantum Gas, Phys. Rev. Lett., 96, 130602, (2006). [5] Brack M., Murthy M.V., Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions, J. Phys. A: Math. Gen. 36, 1111–1133, (2003). [6] Bruun G.M., Clark C.W., Ideal gases in time-dependent traps, Phys. Rev. A, 61, 061601(R), (2000). [7] B¨ uttiker M., Prêtre A., Thomas H., Phys. Rev. Lett. 70, 4114, (1993). [8] Butts D.A., Rokhsar D.S., Trapped Fermi Gases, Phys. Rev. A, 55, 4346–4350, (1997). [9] Combescure M., Robert D., Rigorous semiclassical results for the magnetic response of an electron gas, Rev. Math. Phys., 13, 1055–1073, (2001). [10] Combescure M., Robert D., Semiclassical results in the linear response theory, Ann. Phys., 305, 45–59, (2003).

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[11] Eckhardt B., Eigenvalue Statistics in Quantum Ideal Gases, arXiv: chao-dyn/ 9809005, (1998). [12] Fock V., Z. Physik, 47, 446–450, (1928). [13] Gat O., Avron J., Semiclassical Analysis and the Magnetization of the Hofstadter Model, Phys. Rev. Lett. 91, 186801, (2003). [14] Gat O., Avron J., Magnetic fingerprints of fractal spectra and the duality of Hofstadter models, New J. Phys., 5, 44.1–44.8, (2003). [15] Gleisberg F., Wonneberger W., Noninteracting fermions in a one-dimensional harmonic atom trap: Exact one-particle properties at zero temperature, Phys. Rev. A 62, 063602, (2000). [16] Helffer B., Sj¨ ostrand J., On diamagnetism and the de Haas-Van Alphen effect, Ann. I. H.P., Phys. Théor., 52, 303–352, (1990). [17] Kubasiak A., Korbicz J., Zakrzewski J., Lewenstein M., Fermi-Dirac statistics and the number theory, Europhysics Letters, (2005). [18] Landau L., Z. Physik, 64, 629, (1930). [19] Leboeuf P. Monastra A., Quantum thermodynamic fluctuations of a chaotic Fermigas model, Nuclear Physics A, 724, 69–84, (2003). [20] Lert P.W., Weare J.H., Static semiclassical response of a bounded electron gas. 1, J. Chem. Phys., 68, 2221–2227, (1978). [21] Levy L.P., Reich D.H., Pfeiffer L., West K., Physica B 189, 204, (1993). [22] Li M., Yan Z., Chen J., Chen L., Chen C., Thermodynamic properties of an ideal Fermi gas in an external potential with U = br t in any dimensional space, Phys. Rev. A 58, 1445–1449, (1998). [23] Molinari V., Sumini M., Rocchi F., Fermion gases in magnetic fields: a semiclassical treatment, Eur. Phys. J. D, 12, 211–217, (2000). [24] Molinari V., Rocchi F., Sumini M., Kinetic Description of Rotating Gases in External Magnetic Fields in the Framework of the Thomas-Fermi-Dirac Approach, Transport Theory and Stat. Phys., 32, 607–621, (2003). [25] Peierls R.E., Z. Physik, 80, 763–791, (1933). [26] Pezzè L., Pitaevskii L., Smerzi A., Stringari S., Insulating Behaviour of a Trapped Ideal fermi Gas, Phys. Rev. Lett. 93, 120401, (2004). [27] Quang D.N., Tung N.H., Semiclassical approach to the density of states of the disordered electron gas in a quantum wire, Phys. Rev. B 60, 13648–136358, (1999). [28] Richter K., Ullmo D., Jalabert R., Orbital magnetism in the ballistic regime: geometrical effects, Phys. Rep. 276, 1–83, (1996). [29] Salasnich L., Ideal quantum gases in D-dimensional space and power-law potentials, Journ. Math. Phys., 41, 8016–8024, (2000). [30] Switkes M., Marcus C.M., Campman K., Gossard A.G., Science, 283, 1907, (1999). [31] Tran M.N., Exact ground-state number fluctuations of trapped ideal and interacting fermions, J. Phys. A: Math. Gen. 36, 961–973, (2003). [32] Tran M.N., Murthy M.V., Bhaduri R.K., Ground-state fluctuations in finite Fermi systems, Phys. Rev. E, 63 031105, (2001). [33] van Faassen E., Dielectric response of a nondegenerate electron gas in semiconductor nanocrystallites, Phys. Rev. B, 58, 15729–15735, (1998).


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[34] van Zyl B.P., Analytical expression for the first-order density matrix of a ddimensional harmonically confined Fermi gas at finite temperature, Phys. Rev. A, 68, 033601, (2003) [35] Yajima K., Existence of solutions for Schr¨ odinger evolution equations, Commun. Math. Phys., 110, 415–426, (1987). Monique Combescure IPNL, Bˆ atiment Paul Dirac 4 rue Enrico Fermi,Université Lyon-1 F-69622 Villeurbanne Cedex, France e-mail: [email protected]


Quadratic Hamiltonians and Their Renormalization Jan Derezi´ nski Abstract. Quadratic bosonic Hamiltonians in Fock spaces with an infinite number of degrees of freedom have a surprisingly rich mathematical theory. In this article I review recent results about two classes of such operators: the van Hove Hamiltonians and the Bogoliubov Hamiltonians. Mathematics Subject Classification (2000). 47B99, 81T99. Keywords. Second quantization, Fock spaces, quadratic Hamiltonians.

1. Introduction In this article I would like to review the results of two recent papers [D, BD] on quadratic bosonic Hamiltonians with an infinite number of degrees of freedom. I would like to convince the reader that their theory is surprisingly rich. Let a∗ξ /aξ denote creation/annihilation operators satisfying [aξ , aξ′ ] = [a∗ξ , a∗ξ′ ] = 0,

[aξ , a∗ξ′ ] = δξ,ξ′ ,

(1)

and acting on a bosonic Fock space. (Above, δξ,ξ′ denotes the delta function. Strictly speaking, a∗ξ /aξ are operator-valued measures and they acquire the meaning of an operator only after smearing out with appropriate test functions.) The first class of Hamiltonians that I would like to discuss was studied in [D] and is given by a formal expression of the form (2) H = h(ξ)a∗ξ aξ dξ + z(ξ)aξ dξ + z(ξ)a∗ξ dξ + c, where h(ξ) ≥ 0. Note that in the above expression the constant c can be infinite. Following [Sch], operators of the form (2) will be called van Hove Hamiltonians. The second class of operators was recently studied by L. Bruneau together with myself in [BD]. Operators from this class are given by a formal expression of

90

J. Derezi´ nski

the form H=

h(ξ)a∗ξ aξ dξ +

1 2

g(ξ, ξ ′ )a∗ξ a∗ξ′ dξ +

1 2

g(ξ, ξ ′ )aξ aξ′ dξ + c.

(3)

We will call them Bogoliubov Hamiltonians. This name is justified by the famous application of such Hamiltonians in the study of the Bose gas due to Bogoliubov [Bog]. There are several questions that one can pose about these operators. They include: When the above formal expressions defines a self-adjoint operator? When they are bounded from below? When they have a ground state? What is their scattering theory? Rather complete answers to these questions exist in the case of van Hove Hamiltonians. For Bogoliubov Hamiltonians, the answers are not so complete, but still we have a number of interesting results about them. Van Hove and Bogoliubov Hamiltonians are used in quantum physics very often. A lot of interesting physical phenomena can be explained just with help of quadratic Hamiltonians. In my paper I would like to convince the reader that also from the mathematical point of view they are interesting objects and illustrate various curious properties of unbounded operators. Quadratic Hamiltonians are also useful, because they help to understand properties of more complicated Hamiltonians used in quantum theory such as those studied in [Fr, DG1, BFS, DJ].

2. Notation Let us briefly review the notation for bosonic Fock spaces that we will use in our paper [Be, RS2, BR, GJ, DG1, D1]. Suppose that Z is a Hilbert space. The bosonic Fock space over the one-particle space Z is defined as ∞

Γs (Z) := ⊕ ⊗ns Z. n=0

It has a distinguished vector called the vacuum vector Ω = 1 ∈ ⊗0s Z = C. The bosonic Fock space can be viewed as a commutative algebra with the product defined as follows: if Ψ ∈ ⊗ns Z, Φ ∈ ⊗m s Z, then Z, Ψ ⊗s Φ := Θs Ψ⊗Φ ∈ ⊗n+m s where Θs is the symmetrizing operator. For z ∈ Z we define the creation operator √ a∗ (z)Ψ := n + 1z ⊗s Ψ, Ψ ∈ ⊗ns Z, ∗

and the annihilation operator a(z) := (a∗ (z)) .

Quadratic Hamiltonians

91

In a large part of the literature one assumes that Z equals L2 (Ξ) for some measure space (Ξ, dξ). One introduces “operator-valued measures” aξ /a∗ξ satisfying (1). If ξ → z(ξ) is a square integrable function then a∗ (z) = z(ξ)a∗ξ dξ, a(z) = z(ξ)aξ dξ.

We will use both notations. The notation involving the operator-valued measures will be called “traditional” – it is lengthy and depends on an arbitrary identification Z = L2 (Ξ), but is perhaps more familiar to some readers and often convenient. For an operator q on Z we define the operator Γ(q) on Γs (Z) by Γ(q) n = q ⊗ · · · ⊗ q. ⊗s Z

For an operator h on Z we define the operator dΓ(h) on Γs (Z) by dΓ(h) n = h ⊗ 1(n−1)⊗ + · · · 1(n−1)⊗ ⊗ h. ⊗s Z

If h is the multiplication operator by h(ξ), then in the traditional notation we have dΓ(h) = h(ξ)a∗ξ aξ dξ. Note the identity Γ(eith ) = eitdΓ(h) . For g ∈ ⊗2s Z we define the 2-particle creation operator a∗ (g)Ψ := (n + 2)(n + 1)g ⊗s Ψ, Ψ ∈ ⊗ns Z,

and the annihilation operator a(g) = a∗ (g)∗ . In the traditional notation, if g equals the function g(ξ, ξ ′ ), then we have a∗ (g) = g(ξ, ξ ′ )a∗ξ a∗ξ′ dξdξ ′ , a(g) = g(ξ, ξ ′ )aξ aξ′ dξdξ ′ .

3. Van Hove Hamiltonians In this section we summarize properties of van Hove Hamiltonians, following [D]. Let Z = L2 (Ξ). Let Ξ ∋ ξ → h(ξ) be a positive function. Let ξ → z(ξ) be a function on Ξ such that |z(ξ)|2 dξ < ∞. |z(ξ)|2 dξ + 2 h≥1 h(ξ) h 0, then (−(ω + ρ2 ), (ω + ρ2 )) is a gap in the essential spectrum containing at most 2N + 1 eigenvalues. Proof. As in the proof of the previous theorem, choose ψ ∈ DC such that E(0, ∞)ψ = ψ

(or E(−∞, 0)ψ = ψ).

Then if ψn = ψ, φn it follows that ∞

2 ψ2n =

n=1

For such a vector ψ it is also true that E(0, ∞)Cψ = Cψ

∞

2 ψ2n−1 .

n=1

(or E(−∞, 0)Cψ = Cψ).

Jacobi Matrices: Eigenvalues and Spectral Gaps

109

Thus 1 2 1 C ψ, ψ = Cψ, Cψ 2 2 ∞ = |a2k−1 ψ2k−1 + a2k ψ2k+1 |2 ≥ = =

k=1 ∞

k=N ∞

|a2k−1 ψ2k−1 + a2k ψ2k+1 |2 |((2k − 1)α + c2k−1 )ψ2k−1 + ((2k)α + c2k )ψ2k+1 |2

k=N ∞

|((2k − 1)α + c2k−1 )ψ2k−1 + ((2k)α + c2k−1 )ψ2k+1

∞

|((2k − 1)α + c2k−1 )ψ2k−1 + ((2k)α + c2k−1 )ψ2k+1 |2

k=N

+ (c2k − c2k−1 )ψ2k+1 |2

=

k=N

+

∞

k=N

2(c2k − c2k−1 )[((2k − 1)α + c2k−1 )ψ2k−1

+ ((2k)α + c2k−1 )ψ2k+1 ]ψ2k+1 +

∞

k=N

2 (c2k − c2k−1 )2 ψ2k+1 .

side will now be viewed as a sum S1 + S2 , where S1 = ∞ The right-hand α α 2 k=N |((2k − 1) + c2k−1 )ψ2k−1 + ((2k) + c2k−1 )ψ2k+1 | . Lower bounds will be found for each of these two pieces which will lead to the stated result.

∞ ((2k − 1)α + c2k−1 )ψ2k−1 + (2k)α + c2k+1 + c2k−1 ψ2k+1 S1 = 2 k=N 2

c2k−1 − c2k+1 ψ2k+1 + 2 2

∞ ((2k − 1)α + c2k−1 )ψ2k−1 + (2k)α + c2k−1 + c2k+1 ψ2k+1 = 2 k=N

∞ c2k−1 − c2k+1 [((2k − 1)α + c2k−1 )ψ2k−1 2 k=N

c2k−1 + c2k+1 ψ2k+1 ]ψ2k+1 + (2k)α + 2 ∞ (c2k−1 − c2k+1 )2 2 + ψ2k+1 4

+2

k=N

110

J. Dombrowski ≥2

# $ ∞ c2k−1 − c2k+1 c2k−1 + c2k+1 2 (2k)α + ψ2k+1 2 2 k=N

∞ c2k−1 − c2k+1 1 2 1 2 ((2k − 1)α + c2k−1 ) ψ2k−1 + ψ2k+1 −2 2 2 2 k=N

∞ (c2k−1 − c2k+1 )2 2 ψ2k+1 + 4 k=N

# $ ∞ c2k−1 − c2k+1 1 c2k+1 2 (2k)α − (2k − 1)α + ψ2k+1 ≥2 2 2 2 k=N

∞ c2k−1 − c2k+1 1 2 [(2k − 1)α + c2k−1 ] ψ2k−1 −2 2 2 k=N

∞ (c2k−1 − c2k+1 )2 2 ψ2k+1 + 4 k=N

# $ ∞ c2k−1 − c2k+1 1 c2k+1 2 (2k)α − (2k − 1)α + ψ2k+1 ≥2 2 2 2 k=N

∞ c2k+1 − c2k+3 1 2 [(2k + 1)α + c2k+1 ] ψ2k+1 −2 2 2 k=N −1

∞ (c2k−1 − c2k+1 )2 2 ψ2k+1 . + 4 k=N

Using the fact that {c2n−1 − c2n+1 } is a non-negative decreasing sequence, it follows that

# $ ∞ 1 c2k−1 − c2k+1 1 2 (2k)α − (2k − 1) − (2k + 1)α ψ2k+1 S1 ≥ 2 2 2 2 k=N

1 2 − (c2N −1 − c2N +1 )[(2N − 1)α + c2N −1 ]ψ2N −1 2 ∞ 2 (c2k−1 − c2k+1 ) 2 ψ2k+1 . + 4 k=N

The strategy for determining a lower bound for S2 is similar. S2 =

∞

k=N

+

2(c2k − c2k−1 )[((2k − 1)α + c2k−1 )ψ2k−1 + ((2k)α + c2k−1 )ψ2k+1 ]ψ2k+1 ∞

k=N

2 (c2k − c2k−1 )2 ψ2k+1

Jacobi Matrices: Eigenvalues and Spectral Gaps ≥

∞

k=N

2 2(c2k − c2k−1 )((2k)α + c2k−1 )ψ2k+1

−2 + ≥2

∞

k=N

∞

k=N ∞

k=N

(c2k − c2k−1 )((2k − 1)α + c2k−1 )

#

1 2 1 ψ + ψ2 2 2k+1 2 2k−1

$

2 (c2k − c2k−1 )2 ψ2k+1

1 1 2 (c2k − c2k−1 ) (2k)α + c2k−1 − (2k − 1)α ψ2k+1 2 2 ∞

−2 +

111

1 2 (c2k+2 − c2k+1 )((2k + 1)α + c2k+1 ) ψ2k+1 2

k=N −1 ∞

k=N

2 (c2k − c2k−1 )2 ψ2k+1 .

Since it was assumed that {c2n − c2n−1 } is a decreasing sequence for n ≥ N , it follows that 2 S2 ≥ −(c2N − c2N −1 )[(2N − 1)α + c2N −1 ]ψ2N −1

+2

∞

k=N

(c2k − c2k−1 )

1 1 1 1 2 × (2k)α + c2k−1 − (2k − 1)α − (2k + 1)α − c2k+1 ψ2k+1 2 2 2 2 ∞ 2 + (c2k − c2k−1 )2 ψ2k+1 . k=N

Combining the two pieces it follows that if ψ, φ1 = 0, . . . , ψ, φ2N −1 = 0 then C 2 ψ, ψ = Cψ2 ≥ ( ρ2 + ω)2 ψ2 . As in the proof of the previous theorem this argument does not apply to an eigenvector corresponding to a zero eigenvalue (when such an eigenvector exists). Hence it leads to the conclusion that (−(ω + ρ ρ 2 ), (ω + 2 )) is a gap in the essential spectrum of the operator C containing at most 2N + 1 eigenvalues. Corollary 2.6. For 0 < α ≤ 1, let an = nα + cn , n = 1, 2, . . . , where the sequence {cn } is chosen so that an > 0 for all n. Suppose that for n ≥ N , c2n−1 − c2n+1 = ρ with ρ ≥ 0, and c2n −c2n−1 = ω with ω ≥ 0. If ρ > 0 or ω > 0, then (−(ω+ ρ2 ), (ω+ ρ 2 )) is a gap in the essential spectrum containing at most 2N+1 eigenvalues.

112

J. Dombrowski

3. Absolute continuity This section presents some results on absolute continuity in the special case α = 1. Theorem 3.1. Let an = n + cn , n = 1, 2, . . . where the sequence {cn } is chosen so that for all n, an > 0, for n ≥ 2, c2n − c2n−2 = ρ, ρ ≥ 0, and for n ≥ 1, c2n−1 − c2n = ω, 0 ≤ ω < 1. 1. If c2 −ρ+ 12 (1−ω) > 0, then (−ω − ρ2 , ω+ ρ2 ) is a gap in the essential spectrum and the spectral˜measure is absolutely continuous on (−∞, 0) ∪ (0, ∞). 2. If 1 + c2 < ρ2 , then (−ω − ρ2 , ω + ρ2 ) is a gap in the essential spectrum and the spectral measure is absolutely continuous on (−∞, −ω − ρ2 ) ∪ (ω + ρ2 , ∞). There exists a non-zero eigenvalue in the spectral gap.

Proof. Under these assumptions, the existence of the spectral gap follows from Theorem 2.1 above. In addition, using the notation from Theorem 2.1 in [6], a = a1 = 1 + c1 > 0, and for n = 1, 2, . . . . δ = a2n − a2n−1 = 1 − ω,

d = a2n+1 − a2n = 1 + c2n+1 − c2n+2 + c2n+2 − c2n = 1 + ρ + ω. Since ω = c1 − c2 , it follows that a1 = 1 + ω + c2 . If c2 − ρ + 12 (1 − ω) > 0 then a − d + 2δ > 0 and it follows from Theorem 2.1 in [6] that the spectral measure is absolutely continuous on (−∞, 0) ∪ (0, ∞). This establishes (i). To prove (ii), let a = a1 = r(ω + ρ2 ), r > 0. Then from Theorem 2.1 in [6] it is sufficient to show that a − d +

δ 2

+

ρ 2 d (ω+ 2 ) > 2 · a2 ρ 2 δ d (ω+ 2 ) + · 2 2 2 a

0.

But a − d + = r(ω + ρ2 ) − (1 + ω + ρ) + 21 (1 − ω) + 2r12 (1 + ω + ρ). Viewing the right-hand side as a function of r, it can be shown that the minimum 1 ρ 3 occurs when r = [ 1+ρ+ω ω+ ρ ] > 1. If 1 + c2 < 2 , then 0 < r < 1, and in this case (ω+ ρ )2

2

a − d + 2δ + d2 · a22 > 0. This establishes absolute continuity outside the spectral gap. It was shown in [6] that zero is an eigenvalue if and only if δ > d. For the case in question zero is not an eigenvalue. Note that λ = a1 = 1 + c1 = 1 + ω + c2 is a root for P3 (λ). If s(λ) = {Pn (λ)} then the modified Wronskian Wn (s(λ), s(−λ)) = (−1)n 2an Pn (λ)Pn (−λ) has at least one node, and it follows from results in [11] that the interval (−λ, λ) contains a non-zero eigenvalue. Theorem 3.2. Let an = n + cn , n = 1, 2, . . . where the sequence {cn } is chosen so that for all n an > 0, c2n−1 − c2n+1 = ρ with ρ ≥ 0, c2n − c2n−1 = ω with 0 ≤ ω ≤ 1 − ρ. If c1 + ρ + 32 ω + 12 > 0 then (−ω − ρ2 , ω + ρ2 ) is a gap in the essential spectrum, and the spectral measure is absolutely continuous on (−∞, 0) ∪ (0, ∞). Proof. It follows from Theorem 2.5 above that (−ω − ρ2 , ω + ρ2 ) is a gap in the essential spectrum containing at most 3 eigenvalues. Theorem 2.1 in [6] again leads to a result on absolute continuity. Using the notation from Theorem 2.1

Jacobi Matrices: Eigenvalues and Spectral Gaps

113

in [6], a = a1 = 1 + c1 > 0, and for n = 1, 2, . . . . δ = a2n − a2n−1 = 1 + ω,

d = a2n+1 − a2n = 1 + c2n+1 − c2n−1 + c2n−1 − c2n = 1 − ρ − ω.

It follows that 1 δ 1 3 a − d + = (1 + c1 ) − (1 − ω − ρ) + (1 + ω) = c1 + ρ + ω + > 0. 2 2 2 2 Thus it follows from Theorem 2.1 in [6] that the spectral measure is absolutely continuous on (−∞, 0) ∪ (0, ∞).

References [1] J. Berezanskii, Expansions in Eigenfunctions of Self-adjoint Operators, Trans. Math. Monographs. 17 Amer. Math. Soc. Providence, RI 1968. [2] J. Dombrowski, Cyclic operators, commutators and absolutely continuous measures, 100 (1987), 457–463. [3] J. Dombrowski, Eigenvalues and spectral gaps related to periodic perturbations of Jacobi matrices, Operator Theory: Advances and Applications, 154 (2004), 91–100. [4] J. Dombrowski and S. Pedersen, Orthogonal polynomials, spectral measures and absolute continuity, J. Comput. Appl. Math., 65 (1995), 115–124. [5] J. Dombrowski and S. Pedersen, Spectral measures and Jacobi matrices related to Laguerre type systems of orthogonal polynomials, Constr. Approx., 13 (1997), 421– 433. [6] J. Dombrowski and S. Pedersen, Spectral Transition parameters for a class of Jacobi matrices, Studia Math., 152 (2002), 217–229. [7] J. Dombrowski, J. Janas. M. Moszynski and S. Pedersen, Spectral gaps resulting from periodic perturbations of a class of Jacobi operators, Constr. Approx., 20 (2004), 585–601. [8] J. Janas, S. Naboko and G. Stolz, Spectral theory for a class of periodically perturbed unbounded Jacobi matrices: elementary methods, J. Comput. Appl. Math., 171 (2004), 265–276. [9] M. Moszynski, Spectral properties of some Jacobi matrices with double weights, J. Math. Anal. Appl., 280 (2003), 400–412. [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [11] G. Teschl, Jacobi Operators and Complete Integrable Nonlinear Lattices, Amer. Math. Soc., Providence, R.I., 2000. J. Dombrowski Department of Mathematics and Statistics Wright State University Dayton, OH 45435, USA e-mail: [email protected]


Eigenvalue Estimates for the Aharonov-Bohm Operator in a Domain Rupert L. Frank and Anders M. Hansson Abstract. We prove semi-classical estimates on moments of eigenvalues of the Aharonov-Bohm operator in bounded two-dimensional domains. Moreover, we present a counterexample to the generalized diamagnetic inequality which was proposed by Erd˝ os, Loss and Vougalter. Numerical studies complement these results. Mathematics Subject Classification (2000). Primary 35P15; Secondary 35J10, 81Q10. Keywords. Spectral estimates, Berezin-Li-Yau inequalities, Aharonov-Bohm magnetic field.

1. Introduction We shall study inequalities for the eigenvalues of the Aharonov-Bohm operator HαΩ := (D − αA0 )2

in L2 (Ω).

(1.1)

Here Ω ⊂ R is a bounded domain, D := −i∇ and αA0 (x) := α|x| (−x2 , x1 )T is a vector potential generating an Aharonov-Bohm magnetic field with flux α through the origin. We shall assume that this point belongs to the interior of the simply-connected hull of Ω and that α ∈ Z, for otherwise αA0 can be gauged away. On the boundary of Ω we impose Dirichlet boundary conditions. More precisely, the operator (1.1) is defined through the closure of the quadratic form (D − αA0 )u2 on C0∞ (Ω \ {0}). Before stating our main results we would like to recall some well-known semiclassical spectral asymptotics and estimates for the Dirichlet Laplacian −∆Ω and its magnetic version (D − A)2Ω , A being an arbitrary vector potential. If Ω is bounded then the spectrum of −∆Ω is discrete, and by a classical result due to 2

−2

116

R.L. Frank and A.M. Hansson

Weyl (see, e.g., [ReS]) one has, as Λ → ∞, 1 1 tr(−∆Ω − Λ)γ− ∼ (|ξ|2 − Λ)γ− dx dξ = |Ω|Λγ+1 2 (2π) 4π(γ + 1) Ω×R2

(1.2)

for all γ ≥ 0. Note that the right-hand side involves the symbol |ξ|2 on the phase space Ω × R2 . The asymptotics (1.2) are accompanied by the estimate 1 (|ξ|2 − Λ)γ− dx dξ, γ ≥ 0, (1.3) tr(−∆Ω − Λ)γ− ≤ Rγ 2 (2π) Ω×R2 with a universal constant Rγ independent of Ω and Λ, and one is interested in the sharp value of this constant Rγ . In view of (1.2) the sharp constant obviously cannot be smaller than 1, and by an argument of Aizenman and Lieb [AL], it is a non-increasing function of γ. P´ olya [P] proved the estimate (1.3) for γ = 0 with constant 1 under the additional assumption that Ω is a tiling domain. His famous conjecture that this is true for arbitrary domains is still unproved. Berezin [B1] and independently Li and Yau [LiY] (see also [La]) proved (1.3) for γ ≥ 1 with the sharp constant Rγ = 1. This also yields the so far best-known bound on the sharp constant for γ = 0, namely R0 ≤ 2. Indeed, 1 |Ω|µ2 , µ > Λ, (1.4) 8π and the claim follows by optimization with respect to µ. We note that the estimate (1.3) for γ = 0 and γ = 1 is closely related to the estimates tr(−∆Ω − Λ)0− ≤ (µ − Λ)−1 tr(−∆Ω − µ)− ≤ (µ − Λ)−1

−1 N λΩ N ≥ ρ0 4π|Ω|

and

N j=1

−1 2 λΩ N j ≥ ρ1 2π|Ω|

Ω for the eigenvalues λΩ j of the operator −∆ . The form (1.3), however, shows the close connection with the Lieb-Thirring inequality, see [LT] and also the review article [LaW2]. We now turn to the ‘magnetic’ analog of (1.3), i.e., where −∆Ω is replaced by the Dirichlet realization of the operator (D − A)2Ω in L2 (Ω) and A is a (sufficiently regular) magnetic vector potential. Note that the value of the right-hand side in (1.3) does not change if ξ is replaced by ξ − A(x). Hence one is interested in the estimate 1 tr((D − A)2Ω − Λ)γ− ≤ Rγmag (|ξ|2 − Λ)γ− dx dξ, γ ≥ 0, (1.5) 2 (2π) Ω×R2

with a universal constant Rγmag independent of Ω, Λ and A. It is a consequence of the sharp Lieb-Thirring inequality by Laptev and Weidl [LaW1] that Rγmag = 1 for γ ≥ 23 . Not much is known about (1.5) in the case γ < 23 . The Laptev-Weidl result and an argument similar to (1.4) yield the (probably non-sharp) estimate Rγmag ≤ ( 53 )3/2 (γ/(γ + 1))γ for γ < 23 . For γ = 0 and γ = 1 in particular one finds the values 2.1517 and 1.0758, respectively. In [ELoV] the estimate (1.5) is shown

Eigenvalue Estimates for the Aharonov-Bohm Operator

117

to hold for γ ≥ 1 with constant 1 in the special case of a homogeneous magnetic field B (1.6) A(x) = (−x2 , x1 )T . 2 It was recently shown in [FLoW] that (1.5) does not hold with constant 1 if 0 ≤ γ < 1, not even when Ω is tiling. Moreover, the authors determined the optimal constant such that (1.5) holds for all Λ and all Ω under the constraint that A is given by (1.6). In this paper we shall consider A corresponding to an Aharonov-Bohm magnetic field and we shall prove the estimate 1 (|ξ|2 − Λ)γ− dx dξ, γ ≥ 1, (1.7) tr(HαΩ − Λ)γ− ≤ Cγ (α) (2π)2 Ω×R2 with a constant Cγ (α) given explicitly in terms of Bessel functions. Even though our bound is probably not sharp, it improves upon the previously known estimates. Indeed, numerical evaluation of our constant shows that (1.7) holds for all α with constants C0 (α) = 1.0540 and C1 (α) = 1.0224 if γ = 0 and 1, respectively, see Section 4. We complement our analytical results with a numerical study of the eigenvalue of the operator (1.1) for five domains: a disc, a square and three different annuli. In all cases the estimate (1.7) seems to be valid with constant 1. We refer to Section 5 and Figures 1–10 for a detailed account of the outcome of our experiments. For the proof of our eigenvalue estimate we proceed similarly as in [ELoV]. Indeed, by the Berezin-Lieb inequality, (1.7) is an immediate consequence of the generalized diamagnetic inequality tr χΩ (Hα −Λ)γ− χΩ ≤ Rγ (α) tr χΩ (−∆−Λ)γ− χΩ 2

for all bounded Ω ⊂ R2 . (1.8)

Here Hα := HαR denotes the Aharonov-Bohm operator in the whole space. In [ELoV] an analogous estimate was proved in the case (1.6) with constant 1 when γ ≥ 1. The authors conjectured that such an inequality is not true for an arbitrary magnetic field, but their counterexample contains a gap; the condition 2) on p. 905 can not be satisfied by a non-trivial radial vector field, as it was first pointed out by M. Solomyak. This gap can be removed by a minor change in the argument, since the assumption of radial symmetry is not essential in the proof [ELo]. In the present paper we show that (1.8) does not hold with constant 1 in the case of the Aharonov-Bohm operator. We feel that our example is of independent interest and sheds some light on the particularities of the Aharonov-Bohm operator. We even calculate the sharp constant in (1.8), see Theorem 3.1. We establish this by a thorough study of the local spectral density of the operator Hα , see Section 2. We mention in closing the papers [BaEvLe], [EkF], [H], [MOR], where LiebThirring estimates for the Schr¨ odinger operator Hα + V were obtained. Our estimates can be seen as a refinement of these estimates in the special case where the potential V is equal to a negative constant −Λ inside and equal to infinity outside a bounded domain Ω ⊂ R2 .

118


2. The Aharonov-Bohm operator in the whole space 2.1. Diagonalization In this section we recall some well-known facts about the Aharonov-Bohm operator in the whole space, see, e.g., [AhBo], [Ru]. We denote by Hα the self-adjoint operator in L2 (R2 ) associated with the closure of the quadratic form |(D − αA0 )u|2 dx, u ∈ C0∞ (R2 \ {0}). R2

Here D := −i∇, A0 (x) := |x|−2 (−x2 , x1 )T and α ∈ R. Moreover, Jν denotes, as usual, the Bessel function of the first kind of order ν, see [AbSt]. With polar coordinates x = |x|(cos θx , sin θx ) and similarly for ξ, we define 1 Fα (ξ, x) := J|n−α| (|ξ||x|)ein(θξ −θx ) , ξ, x ∈ R2 , 2π n∈Z

and put

(Fα u)(ξ) :=

Rd

Fα (ξ, x)u(x) dx,

ξ ∈ R2 ,

for u ∈ C0∞ (R2 ). Note that F0 (ξ, x) = (2π)−1 e−iξ·x [AbSt, 9.1.41], so F0 is the ordinary Fourier transform, which diagonalizes H0 = −∆. Similarly, one has Lemma 2.1. For any α ∈ R, Fα extends to a unitary operator in L2 (R2 ) and diagonalizes Hα , i.e., (Fα f (Hα )u)(ξ) = f (|ξ|2 )(Fα u)(ξ), for any u ∈ L2 (R2 ) and f ∈ L∞ (R).

ξ ∈ R2 ,

We sketch a proof of this assertion for the sake of completeness. Proof. The orthogonal decomposition L2 (R2 ) = Hn , Hn := {|x|−1/2 g(|x|)einθx : g ∈ L2 (R+ )}, n∈Z

reduces Hα . The part of Hα in Hn is unitarily equivalent to the operator

(n − α)2 − 1/4 d2 + in L2 (R+ ), 2 dr r2 which is defined as the Friedrichs extension of the corresponding differential expression on C0∞ (R+ ). (We emphasize that in our notation R+ means the open interval (0, ∞).) The operator √ rkJν (rk)g(r) dr, k ∈ R+ , (Φν g)(k) := h|n−α| := −

R+

initially defined for g ∈ C0∞ (R+ ), extends to a unitary operator in L2 (R+ ) and diagonalizes hν , i.e., (Φν f (hν )g)(k) = f (k 2 )(Φν g)(k),

k ∈ R+ ,


119

for any g ∈ L2 (R+ ) and f ∈ L∞ (R) [Ti, Ch. VIII]. The assertion of the lemma is a simple consequence of these facts. The proof of the preceding lemma shows in particular that the operators Hα and Hα+m with m ∈ Z are unitarily equivalent via multiplication by eimθx (a gauge transformation). Hence, without loss of generality we shall assume that 0 ≤ α < 1. Lemma 2.1 implies that f (Hα ), at least formally, is an integral operator with integral kernel f (Hα )(x, y) = Fα (ξ, x)f (|ξ|2 )Fα (ξ, y) dξ R2 1 ∞ = f (k 2 )J|n−α| (k|x|)J|n−α| (k|y|)ein(θx −θy ) k dk 2π 0 n∈Z √ √ 1 ∞ f (λ)J|n−α| ( λ|x|)J|n−α| ( λ|y|)ein(θx −θy ) dλ. = 4π 0 n∈Z

On the diagonal this is

f (Hα )(x, x) = where ρα (t) :=

1 4π

n∈Z

∞

√ f (λ)ρα ( λ|x|) dλ,

(2.1)

0

2 J|n−α| (t),

t ≥ 0.

(2.2)

√ 1 In particular, 4π ρα ( λ|x|) is the local spectral density at energy λ. In the following subsection we collect some basic information about this function, and in Subsections 2.3 and 2.5 we prove some results about the precise asymptotic behavior of (2.1) as |x| → ∞ for special choices of functions f . This will allow us to prove that the generalized diamagnetic inequality is violated. 2.2. The spectral density Our results are based on a detailed study of the function ρα from (2.2). We note that if α = 0 then ρ0 ≡ 1 by [AbSt, 9.1.76]. An expression in terms of elementary functions is also available [AbSt, 5.2.15] for α = 1/2, 2 2t sin s ds. (2.3) ρ1/2 (t) = π 0 s As t → ∞, ρ1/2 (t) tends to 1 in an oscillating manner. As we shall see, this behavior appears for all 0 < α < 1. The starting point of our study of ρα with non-trivial flux α is the following

Lemma 2.2. For any 0 < α < 1, ρα is a smooth function on R+ with ρα (0) = 0, ρα (t) → 1 as t → ∞ and ρ′α (t) = Jα (t)Jα−1 (t) + J1−α (t)J−α (t),

t ≥ 0.

(2.4)

120


Proof. By [Lu, 11.2(10)] and [AbSt, 11.4.42] one has, for all t ≥ 0, t ρα (t) = (Jα (s)Jα−1 (s) + J1−α (s)J−α (s)) ds 0 ∞ =1− (Jα (s)Jα−1 (s) + J1−α (s)J−α (s)) ds, t

which implies the assertion.

Our next result will not be needed in the sequel, but it helps to clarify the behavior of ρα and demonstrates the methods which we shall use later on. Lemma 2.3. Let 0 < α < 1. As t → ∞, ρα (t) = 1 −

sin απ cos 2t + O(t−2 ). π t

Proof. The asymptotics [AbSt, 9.2.5] 8

π νπ . 4ν 2 − 1 π νπ . 2 cos t − − − + O(t−2 ) , sin t − − Jν (t) = πt 4 2 8t 4 2

the formula (2.4) and elementary manipulations show that

(2α − 1)2 sin απ cos 2t 2 sin απ sin 2t + ρ′α (t) = + O(t−2 ) . πt 4 t

∞ ′ Using that ρα (t) = 1 − t ρα (s) ds by Lemma 2.2, we obtain the assertion by repeated integration by parts. 2.3. Moments of the spectral density For any γ > −1 let us define 1 √ (1 − λ)γ ρα ( λr)dλ, σα,γ (r) := 0

r ≥ 0.

(2.5)

We are interested in the asymptotic behavior of this quantity or, more precisely, in the way it approaches its limit. Theorem 2.4. Let 0 < α < 1 and γ > −1. As r → ∞, σα,γ (r) =

sin απ sin(2r − 21 γπ) 1 − Γ(γ + 1) + O(r−3−γ ). γ+1 π r2+γ

For the proof we note that by Lemma 2.2 and dominated convergence 1 1 . lim σα,γ (r) = (1 − λ)γ dλ = r→∞ γ + 1 0

In view of

σα,γ (r) =

1 − γ+1

r

∞

′ σα,γ (s) ds,


121

Theorem 2.4 follows via integration by parts from Proposition 2.5. Let 0 < α < 1 and γ > −1. As r → ∞,

γπ . d1 γπ . Γ(γ + 1) sin απ ′ −2 cos 2r − sin 2r − − + O(r−2 ) σα,γ (r) = γ+2 r π 2 r 2 (2.6) 3 2 1 2 where d1 := 8 (1 − 4((γ + 2 ) + (1 − 2α) ). We defer the rather technical proof of this proposition to the following subsection. We would like to point out that (2.6) can be proved in an elementary way if α = 21 due to the formula (2.3). Indeed, if we simplify further by taking γ = 1, then 1 1 √ 2 4 ′ σ1/2,1 (1 − λ) sin(2 λr) dλ = k(1 − k 2 ) sin 2kr dk (r) = πr 0 πr 0

2 sin 2r 3 cos 2r =− + + O(r−4 ) 2 3 πr r 2r by integration by parts. This is (2.6) in this special case. 2.4. Proof of Proposition 2.5 We shall need Lemma 2.6. Let ν > −1/2 and γ > −1. Then 1 Jν+1/2 (kr)Jν−1/2 (kr)k 2 (1 − k 2 )γ dk = cγ 0

0

1

J2ν (2kr)k(1 − k 2 )γ+1/2 dk

with 1 Γ(γ + 1) . cγ := √ π Γ(γ + 32 )

(2.7)

Proof. By virtue of the series representations [AbSt, 9.1.10, 9.1.14] for α > 0, 2n α−1 ∞ (−1)n t t Jα−1 (t) = , 2 n!Γ(α + n) 2 n=0 2n 2α−1 ∞ t (−1)n Γ(2α + 2n) t Jα (t)Jα−1 (t) = , 2 n!Γ(α + n)Γ(α + n + 1)Γ(2α + n) 2 n=0

the statement is equivalent to

1 (−1)n (r/2)2n Γ(2ν + 2n) k 2ν+2n+1 (1 − k 2 )γ dk n!Γ(2ν + n)Γ(ν + n)Γ(ν + n + 1) 0 n=0 1 ∞ r2ν−1 Γ(γ + 1) (−1)n r2n = √ k 2ν+2n (1 − k 2 )γ+1/2 dk, (2.8) π Γ(γ + 23 ) n=0 n!Γ(2ν + n) 0

∞ - r .2ν−1

2

122


an equality which actually holds termwise. This follows by the beta function identity 1 1 Γ(α)Γ(β) 2 k 2α−1 (1 − k 2 )β−1 dk = tα−1 (1 − t)β−1 dt = B(α, β) = Γ(α + β) 0 0 √ and the duplication formula πΓ(2α) = 22α−1 Γ(α)Γ α + 12 [AbSt, 6.1.18]. Lemma 2.7. Let β > 0 and −2 < ν < 2. The following decomposition holds: 1 Jν (kr)k(1 − k 2 )β−1 dk = fβ,ν (r) + gβ,ν (r), (2.9) 0

where gβ,−ν (r) = −gβ,ν (r) and β+1/2 # $ d1 Γ(β) 2 sin(r − r0 ) + O(r−2 ) , cos(r − r0 ) + fβ,ν (r) = √ 2 π r r

with r0 :=

π 2

1 β+ν + 2

and

d1 :=

1 β2 + ν2 − . 8 2

r → ∞, (2.10)

The function gβ,ν has a power-like, non-oscillatory behavior at infinity and dominates fβ,ν when β > 32 . Its odd parity with respect to ν is, however, the property vital to us, since it leads to a useful cancellation. Proof. The left-hand side in (2.9) can be expressed as a generalized hypergeometric function. Recall that for p, q ∈ N0 , p ≤ q and α1 , . . . , αp , β1 , . . . , βq ∈ R \ (−N0 ), ∞ (α1 )n (α2 )n · · · (αp )n z n α1 , α2 , . . . , αp , z ∈ C, z = p Fq β1 , β2 , . . . , βq (β1 )n (β2 )n · · · (βq )n n! n=0 where (α)n = Γ(α + n)/Γ(α) = α(α + 1) · · · (α + n − 1). By [Lu, 13.3.2(10)], π/2 1 Jν (kr)k(1 − k 2 )β−1 dk = Jν (r sin θ) cos2β−1 θ sin θ dθ 0 0

r2 rν Γ(β)Γ(ν/2 + 1) ν/2 + 1 − . = ν+1 1 F2 ν + 1, β + ν/2 + 1 2 Γ(ν + 1)Γ(β + ν/2 + 1) 4

Next we use the asymptotics of the generalized hypergeometric function [Lu, 1.3.3(5, 7, 8, 13)],

r2 Γ(ν/2 + 1) ν/2 + 1 − 1 F2 ν + 1, β + ν/2 + 1 Γ(ν + 1)Γ(β + ν/2 + 1) 4 : 9 ∞ ∞ (2/r)β+ν+1/2 i(r−r0 ) −n −n −i(r−r0 ) √ e ∼ dn (−ir) dn (ir) + e 2 π n=0 n=0

ν(2/r)ν+2 4 1 + ν/2, 1 − ν/2, 1 − β + r → ∞, 3 F0 − r2 , − 2Γ(β)


123

with r0 , d1 as in (2.10) and d0 = 1. We emphasize that the ∼ sign means equality in the sense of asymptotic expansions, and that 3 F0 is not a well-defined function but denotes an asymptotic expansion. We define ν −1 r2 gβ,ν (r) as a finite approximation to 3 F0 . Namely, let K denote the smallest non-negative integer such that 2K ≥ β − 32 and put 2n K (−1)n 2 ν gβ,ν (r) := 2 . (1 + ν/2)n (1 − ν/2)n (1 − β)n r n=0 n! r This function is antisymmetric in ν and 1 Jν (kr)k(1 − k 2 )β−1 dk − gβ,ν (r) fβ,ν (r) := 0 #

d1 d1 rν Γ(β) (2/r)β+ν+1/2 i(r−r0 ) √ e + e−i(r−r0 ) d0 − d0 + = ν+1 2 2 π ir ir $ . −2 −2(K+2) + O(r ) + O r Γ(β) = √ 2 π

as claimed.

β+1/2 # $ 2 d1 sin(r − r0 ) + O(r−2 ) d0 cos(r − r0 ) + r r

Now everything is in place for the Proof of Proposition 2.5. By Lemmas 2.2 and 2.6, 1 ′ (r) = 2 k 2 (1 − k 2 )γ ρ′α (kr)dk σα,γ 0

=2

1

0

= 2cγ

(Jα (kr)Jα−1 (kr) + J1−α (kr)J−α (kr))k 2 (1 − k 2 )γ dk

0

1

(J2α−1 (2kr) + J1−2α (2kr))k(1 − k 2 )γ+1/2 dk

with cγ from (2.7). Now we apply Lemma 2.7 with β = γ + 32 and ν = ±(2α − 1). Using the antisymmetry of gβ,ν we find that ′ σα,γ (r) = 2cγ fγ+3/2,2α−1 (r) + fγ+3/2,1−2α (r) ,

and the assertion follows after elementary manipulations from the asymptotics of fβ,ν given in Lemma 2.7. 2.5. The Laplace transform of the spectral density In this subsection we are interested in the quantity ∞ √ σα,∞ (r) := e−λ ρα ( λr)dλ, 0

r ≥ 0.

√ Note that this is essentially the Laplace transform of the function ρα ( ·).

124


Theorem 2.8. For all 0 < α < 1 the function σα,∞ is strictly increasing from 0 to 1 on [0, ∞). In particular, σα,∞ (r) < 1 for all r ≥ 0. This theorem shows that the oscillations we observed for σα,γ are no longer present. Proof. Since ρα = ρ1−α and hence σα,∞ = σ1−α,∞ , it suffices to treat the case 0 < α ≤ 1/2. By the properties of ρα (see Lemma 2.2) and dominated convergence we get σα,∞ (0) = 0 and σα,∞ (r) → 1 as r → ∞. Again by Lemma 2.2, ∞ 2 ′ k 2 e−k ρ′α (kr)dk (r) = 2 σα,∞ 0 ∞ 2 =2 k 2 e−k (Jα (kr)Jα−1 (kr) + J1−α (kr)J−α (kr)) dk, 0

and hence according to [Lu, 13.4.1(10)], # 21−2α α + 21 ′ (r) = r1−2α 2(1−2α) σα,∞ 1 F1 2α r Γ(α) +

− r2

1 1 F1 21−2α Γ(1 − α)

$ − α − r2 . 2 − 2α 3 2

In the special case α = 1/2 we note that

2 2 2 1 2 ′ − r = √ e−r . (r) = √ 1 F1 σ1/2,∞ 1 π π

If 0 < α < 12 we can apply the Kummer transformations [AbSt, 13.1.27] (note that 1 F1 (a, b; z) = M (a, b, z) in [AbSt]) to get # 2 21−2α α − 21 2 ′ r σα,∞ F (r) = r1−2α e−r 1 1 2α r2(1−2α) Γ(α) $ 1 1 2 − α r2 . + 1−2α 1 F1 2 − 2α 2 Γ(1 − α)

By elementary properties of the gamma function this can be rewritten as

2 sin απ 1−2α −r2 1 ′ r − α, 2 − 2α, r2 (r) = √ e U σα,∞ π 2

where [AbSt, 13.1.3] #

1 1 π 1 α − 12 2 r − α, 2 − 2α, r2 = U 1 F1 1 2(1−2α) 2α 2 sin 2απ r Γ( 2 − α)Γ(2α) $ 1 r2(1−2α) 2 − α r2 . − 1 F1 1 2 − 2α Γ(α − 2 )Γ(2 − 2α)

U is positive by the integral representation [AbSt, 13.2.5], and this proves the theorem.


125

3. Counterexample to the generalized diamagnetic inequality Following [ELoV] we consider the question: Which non-negative convex functions ϕ vanishing at infinity satisfy for all bounded domains Ω ⊂ R2 ?

tr χΩ ϕ(Hα )χΩ ≤ tr χΩ ϕ(−∆)χΩ

By (2.1) the statement (3.1) is equivalent to the pointwise inequality ∞ ∞ √ ϕ(λ)ρα ( λr) dλ ≤ ϕ(λ) dλ, for all r ∈ [0, ∞). 0

(3.1)

(3.2)

0

Note that (3.1) is true for the family of functions ϕ(λ) = e−tλ , t > 0; we shall prove this (even with strict inequality) in Remark 3.2 below. Alternatively, it follows from the ‘ordinary’ diamagnetic inequality (see, e.g., [HuS]), | exp(−tHα )u| ≤ exp(−t(−∆))|u| a.e.,

u ∈ L2 (R2 ),

and [S, Thm. 2.13], since ;

;2 ; ; ;χΩ exp − t Hα ; = tr χΩ exp(−tHα )χΩ . ; ; 2 2

We point out that the validity of (3.1) for ϕ(λ) = e−tλ , t > 0, actually implies that it holds for any function of the form ∞ e−tλ w(t) dt, w ≥ 0. (3.3) ϕ(λ) = 0

In connection with a Berezin-Li-Yau-type inequality one is particularly interested in the functions ϕ(λ) = (λ − Λ)γ− ,

γ ≥ 1, Λ > 0.

(3.4)

Note that these functions cannot be expressed in the form (3.3). Theorem 3.1. Let 0 < α < 1 and let ϕ be given by (3.4) for some γ ≥ 1, Λ > 0. Then the generalized diamagnetic inequality (3.1) is violated. More precisely, there exist constants C1 , C2 > 0 (depending on α and γ, but not on Λ) such that for all |x| ≥ C1 Λ−1/2 , √ sin(2 Λ|x| − 12 γπ) Λ(γ−1)/2 (3.5) ϕ(Hα )(x, x) − ϕ(−∆)(x, x) + Aα,γ (Λ) ≤ C2 |x|2+γ |x|3+γ with Aα,γ (Λ) := (2π)−2 Λγ/2 Γ(γ + 1) sin απ.

Proof. By (2.1) and the scaling λ → Λλ, √ √ Λγ+1 1 Λγ+1 σα,γ ( Λ|x|), (1 − λ)γ ρα ( Λλ|x|) dλ = ϕ(Hα )(x, x) = 4π 0 4π

(3.6)

where we used the notation (2.5). Note that ϕ(−∆)(x, x) = (4π(γ + 1))−1 Λγ+1 . The expansion (3.5) is thus a consequence of Theorem 2.4. To prove that the generalized diamagnetic inequality (3.1) is violated, one can consider the domains

126


√ Ωn := {x ∈ R2 : | Λ|x| − rn | < ε}, n ∈ N, with rn := π(n + 14 (γ − 1)) and sufficiently small but fixed ε > 0. It follows easily from (3.5) that (3.1) is violated for all large n. Remark 3.2. The analogous statement (with the same proof) is valid for −1 < γ < 1. Moreover, for exponential functions Theorem 2.8 implies that

∞ √ |x| 1 1 |x| σα,∞ √ ρα λ √ e−λ dλ = exp(−tHα )(x, x) = 4πt 0 4πt t t is strictly less than (4πt)−1 = exp(−t(−∆))(x, x).

The following substitute for (3.1) will be useful later on. Proposition 3.3. Let 0 < α < 1 and let ϕ be given by (3.4) for some γ > −1, Λ > 0. Then for all open sets Ω ⊂ R2 with

tr χΩ ϕ(Hα )χΩ ≤ Rγ (α) tr χΩ ϕ(−∆)χΩ

(3.7)

(3.8)

Rγ (α) := (γ + 1) sup r≥0

1

0

√ (1 − λ)γ ρα ( λr) dλ.

Indeed, this is an immediate consequence of (3.6). This shows as well that the right-hand side of (3.8) yields the sharp constant in (3.7). Remark 3.4. The constant Rγ (α) is strictly decreasing with respect to γ. Indeed, following [AL] we write, for γ > γ ′ > −1 and 0 ≤ λ ≤ 1, 1−λ ′ ′ B(γ − γ ′ , γ ′ + 1)(1 − λ)γ = (1 − λ − µ)γ µγ−γ −1 dµ 0

and find, for any r ≥ 0,

1 √ (1 − λ)γ ρα ( λr) dλ B(γ − γ ′ , γ ′ + 1) 0

1 1−μ √ ′ ′ (1 − λ − µ)γ ρα ( λr) dλ µγ−γ −1 dµ = 0

0

=

0

1

w(r

1 − µ)(1 − µ)γ

′

+1 γ−γ ′ −1

µ

dµ,

√ ′ where w(s) := 0 (1−λ)γ ρα ( λs) dλ. Since w(s) ≤ Rγ ′ (α)(γ ′ +1)−1 one concludes that 1 ′ ′ (1 − µ)γ +1 µγ−γ −1 dµ Rγ (α) ≤ Rγ ′ (α)(γ ′ + 1)−1 (γ + 1)B(γ − γ ′ , γ ′ + 1)−1

1

0

= Rγ ′ (α)(γ ′ + 1)−1 (γ + 1)B(γ − γ ′ , γ ′ + 1)−1 B(γ − γ ′ , γ ′ + 2) = Rγ ′ (α).

That this inequality is actually strict, follows from the fact that the supremum in √ (3.8) is attained for some r0 ∈ (0, ∞) (see Theorem 2.4) and that w(r0 1 − µ) is non-constant with respect to µ.


127

We close this section by giving numerical values for Rγ (α). Note that 1 (1 − k 2 )γ ρα (kr)k dk Rγ (α) =2(γ + 1) sup = sup r r≥0

= sup r r≥0

r≥0

1

0 1 0

0

(1 − k 2 )γ+1 ρ′α (kr) dk (1 − k 2 )γ+1 (Jα (kr)Jα−1 (kr) + J1−α (kr)J−α (kr)) dk.

The integral can be evaluated by some quadrature algorithm and by Theorem 2.4 the supremum is attained for finite r. This allows us to compute approximate values of Rγ (α) (= Rγ (1 − α)), Rγ (0.1) Rγ (0.2) Rγ (0.3) Rγ (0.4) Rγ (0.5)

γ γ γ γ γ

=0 = 21 =1 = 23 =2

1.01682 1.01027 1.00650 1.00417 1.00267

1.03262 1.02050 1.01351 1.00920 1.00642

1.04422 1.02781 1.01833 1.01250 1.00874

1.05151 1.03241 1.02138 1.01457 1.01019

1.05397 1.03395 1.02238 1.01524 1.01065

In addition to the monotonicity in γ (see Remark 3.4) the constant Rγ (α) seems to be strictly increasing in α, but we have not been able to prove this.

4. A magnetic Berezin-Li-Yau inequality As in the introduction, let Ω ⊂ R2 be a bounded domain and define HαΩ in L2 (Ω) through the closure of the quadratic form (D − αA0 )u2 on C0∞ (Ω \ {0}). For eigenvalue moments of this operator we shall prove Theorem 4.1. Let 0 < α < 1, γ ≥ 1 and Ω ⊂ R2 be a bounded domain such that the operator HαΩ has discrete spectrum. Then for any Λ > 0, tr(HαΩ − Λ)γ− ≤ Rγ (α)

1 |Ω|Λγ+1 4π(γ + 1)

(4.1)

with the constant Rγ (α) from (3.8). As explained in the introduction, 1 1 |Ω|Λγ+1 = 4π(γ + 1) (2π)2

R2 ×Ω

(|ξ|2 − Λ)γ− dξ dx

is the semi-classical approximation for tr(HαΩ − Λ)γ− . Unfortunately, we can only prove (4.1) with an excess factor Rγ (α), which is strictly larger than one by Theorem 2.4. Based on numerical calculations (see next section) we conjecture that (4.1) should be valid with Rγ (α) = 1.

128


Proof. We first note that tr(HαΩ − Λ)− ≤ tr χΩ (χΩ Hα χΩ − Λ)− χΩ .

(4.2)

Indeed, let (ωj ) be an orthonormal basis of eigenfunctions of HαΩ . Then the extension ω ˜ j of ωj by zero belongs to the form domain of χΩ Hα χΩ and tr(HαΩ − Λ)− = ((HαΩ − Λ)ωj , ωj )− = ((χΩ Hα χΩ − Λ)˜ ωj , ω ˜ j )− j

≤

j

j

((χΩ Hα χΩ − Λ)− ω ˜j , ω ˜ j ) = tr χΩ (χΩ Hα χΩ − Λ)− χΩ .

Now let ϕ be a convex function of the form ϕ(λ) = w ≥ 0. Then it follows from (4.2) that

(λ − µ)− w(µ) dµ for some

tr ϕ(HαΩ ) ≤ tr χΩ ϕ(χΩ Hα χΩ )χΩ ,

and hence by the Berezin-Lieb inequality (see [B2], [L] and also [LaSa], [La]), tr ϕ(HαΩ ) ≤ tr χΩ ϕ(χΩ Hα χΩ )χΩ ≤ tr χΩ ϕ(Hα )χΩ .

As already noted in Remark 3.4, the function ϕ(λ) = (λ − Λ)γ− is of the considered form. Noting that in this case tr χΩ ϕ(−∆)χΩ = (4π(γ+1))−1 |Ω|Λγ+1 , the assertion follows from Proposition 3.3.

5. Numerical experiments 5.1. Numerical evaluation of the magnetic Berezin-Li-Yau constant In this section we present a numerical study, somewhat in the spirit of [LT, Appendix A], of the constant Rγ (α) appearing in (4.1) for HαΩ on various domains. We consider the unit disc, a square and three annuli: A := {x : |x| < 1},

B := {x : max(|x1 |, |x2 |) < 1},

C := {x : 1 < |x| < 1.1},

D := {x : 1 < |x| < 2},

E := {x : 1 < |x| < 11}.

To expose possible diamagnetic effects we perform identical experiments for α = 0 and α = 0.2 throughout. If the eigenvalues λ1 , λ2 , . . . are known, the task is to determine 4π(γ + 1) (Λ − λj )γ . (5.1) where rγ (Λ) := Rγ = sup rγ (Λ), |Ω|Λγ+1 Λ j: λj 1 (see [H, Thm. 3.2]), namely


129

the Schr¨ odinger operator Hα − |x|−1 . The × symbols on the x axis indicate the eigenvalue loci. 5.2. Computation of the eigenvalues By separation of variables the spectrum of HαA is the union of the spectra of the family of one-dimensional problems (n − α)2 − 1/4 u = λu, u(1) = 0, u ∈ L2 (0, 1), r2 √ √ parametrized by n ∈ Z. rJ|n−α| ( λr) is an eigenfunction provided λ is chosen √ to that J|n−α| ( λ) = 0, i.e., the eigenvalues are squares of the zeros of the Bessel function of the first kind. Using standard numerical procedures for this task we compute all eigenvalues below 10,000 (approximately 2,500) of H0A and all eigenA . The lowest eigenvalues of the two values below 50,000 (approx. 12,000) of H0.2 operators can be viewed in Figure 4. The reader interested in the dependence of the eigenvalues on the parameter α may find Figure 6 useful, where the lowest eigenvalues of HαA as functions of α ∈ [0, 21 ] are shown. We recall that HαA and A share the same spectrum. H1−α Similarly, the eigenvalue problems for HαC , HαD and HαE are reduced to −u′′ +

(n − α)2 − 1/4 u = λu, u(r1 ) = 0 = u(r2 ), u ∈ L2 (r1 , r2 ). r2 √ √ √ In order that the general solution r(c1 J|n−α| ( λr) + c2 Y|n−α| ( λr)) satisfy the Dirichlet conditions we need to fix the ratio c1 /c2 and take λ such that √ √ √ √ J|n−α| ( λr1 )Y|n−α| ( λr2 ) = J|n−α| ( λr2 )Y|n−α| ( λr1 ). −u′′ +

Equivalently, λ = x2 /r12 is an eigenvalue whenever x is a zero of the cross-product Jν (t)Yν (µt) − Jν (µt)Yν (t) for µ = r2 /r1 , ν = |n − α|. Since the amplitude of the oscillation grows approximately as µν /ν, the computation of the zeros is nontrivial for large orders. By searching in the negative t direction and making use of the simple fact that the asymptotic spacing of the zeros is π/(µ − 1) [AbSt, 9.5.28], we circumvent much of these difficulties in comparison to routines such as Mathematica’s BesselJYJYZeros. This allows us to compute, for α = 0 and 0.2, all eigenvalues below 10,000 (approx. 400) of HαC , all eigenvalues below 1,000 (approx. 550) of HαD and all eigenvalues below 300 (approx. 9,000) of HαE . H0B is simply the Dirichlet Laplacian on [−1, 1]2 , so for all k, l ∈ N,

y+1 x+1 sin lπ sin kπ 2 2 2

is an eigenfunction and π4 (k 2 + l2 ) an eigenvalue. For α = 0 the absence of radial symmetry prevents us from solving the eigenvalue problem for HαB exactly. We therefore resort to the finite element method as implemented in Comsol Multiphysics to compute approximate eigenvalues. A 8,800-element mesh, gradually refined near the singularity, is used to compute the 250 lowest eigenvalues, all less

130


than 840. Providing an a priori bound on the computational error would be beyond the scope of this article; it is an encouraging fact, however, that the approximate A do not differ from the exact ones by more than 0.8 %. Figure 5 eigenvalues of H0.2 is the analogue of Figure 4 for HαB . 5.3. Outcome of the experiments We are obviously constrained to computing (5.1) with Λ restricted to a finite interval but provided this interval is sufficiently large it should be possible to tell the behaviors depicted in Figures 2 and 3 apart. Particularly if the function rγ tends steadily to 1 from below we can conclude with certainty that its supremum will eventually be this number in virtue of the eigenvalue asymptotics. It turns out that all the operators we consider, whether α = 0 or 0.2, seem to obey a semi-classical eigenvalue estimate with unit constant for both γ = 0 and 1. What however differs is the rate of convergence to 1, e.g., rγ increases slower for HαC than for HαE since |C| < |E|, implying that the point spectrum of the former operator is sparser. To see the influence of α on rγ we invite the reader to compare Figures 7, 8 and 9. Note that rγ has the most irregular behavior in the cases α = 0 and α = 0.5 due to the high degree of degeneracy of the eigenvalues, cf. Figure 6. Moreover we have the impression, from looking at Figures 1, 7 and 10, that the shape of the domain has a limited influence on rγ . All three plots suggest that R0 = 1 but this has not been proved so far even for H0A , the Laplacian on the unit disc! Acknowledgements The authors would like to thank A. Laptev for the setting of the problem and helpful remarks. The first author is grateful to E.H. Lieb and R. Seiringer for their hospitality at Princeton University and thanks them, H. Kalf and M. Loss for fruitful discussions.


131

γ=1

γ=0 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0

100

50

0

100

50

0

Figure 1. rγ for H0B near the bottom of the spectrum γ=1

γ=0 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

1000

2000

3000

4000

0

1000

2000

Figure 2. rγ (Λ) for H0B for large Λ

3000

4000

132


2

1.5

1

0.5

0 −1.8

−2

−1.6

−1.2

−1.4

−1

−0.8

−0.6

−0.4

−0.2

Figure 3. r0 for an operator with R0 > 1 80

70

60

50

40

30

20

10

0 2

4

6

8

10

12

14

A Figure 4. Lowest eigenvalues of H0A (·) and of H0.2 (×)

0


133

70

60

50

40

30

20

10

0 12

10

8

6

4

2

14

B Figure 5. Lowest eigenvalues of H0B (·) and of H0.2 (×)

60

50

40

30

20

10

0 0

0.05

0.1

0.15

0.2

0.25 α

0.3

0.35

0.4

0.45

Figure 6. Lowest eigenvalues of HαA as function of α

0.5

134

R.L. Frank and A.M. Hansson γ=1

γ=0 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

50

0

100

50

100

Figure 7. rγ for H0A near the bottom of the spectrum γ=1

γ=0 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

50

100

0

50

A Figure 8. rγ for H0.2 near the bottom of the spectrum

100


135

γ=1

γ=0 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

50

0

100

50

100

A Figure 9. rγ for H0.5 near the bottom of the spectrum γ=1

γ=0 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

50

100

0

50

Figure 10. rγ for H0D near the bottom of the spectrum

100

136


References [AhBo]

Y. Aharonov, D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory. Phys. Rev. II 115 (1959), 485–491 [AL] M. Aizenman, E. Lieb, On semiclassical bounds for eigenvalues of Schr¨ odinger operators. Phys. Lett. A 66 (1978), no. 6, 427–429. [AbSt] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, Washington D.C., 1964. [BaEvLe] A.A. Balinsky, W.D. Evans, R.T. Lewis, On the number of negative eigenvalues of Schr¨ odinger operators with an Aharonov-Bohm magnetic field. Proc. R. Soc. Lond. A 457 (2001) 2481–2489 [B1] F.A. Berezin, Covariant and contravariant symbols of operators [Russian]. Math. USSR Izv. 6 (1972), 1117–1151. [B2] F.A. Berezin, Convex functions of operators [Russian]. Mat. Sb. 88 (1972), 268–276. [EkF] T. Ekholm, R.L. Frank, On Lieb-Thirring inequalities for Schr¨ odinger operators with virtual level. Comm. Math. Phys. 264 (2006), no. 3, 725–740. [ELo] L. Erd˝ os, M. Loss, private communication. [ELoV] L. Erd˝ os, M. Loss, V. Vougalter, Diamagnetic behavior of sums of Dirichlet eigenvalues. Ann. Inst. Fourier 50 (2000), 891–907. [FLoW] R.L. Frank, M. Loss, T. Weidl, P´ olya’s conjecture in the presence of a constant magnetic field. J. Eur. Math. Soc., to appear. [H] A.M. Hansson, On the spectrum and eigenfunctions of the Schr¨ odinger operator with Aharonov-Bohm magnetic field. Int. J. Math. Math. Sci. 23 (2005), 3751– 3766. [HuS] D. Hundertmark, B. Simon, A diamagnetic inequality for semigroup differences. J. Reine Angew. Math. 571 (2004), 107–130. [La] A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal. 151 (1997), no. 2, 531–545. [LaSa] A. Laptev, Yu. Safarov, A generalization of the Berezin-Lieb inequality. Amer. Math. Soc. Transl. (2) 175 (1996), 69–79. [LaW1] A. Laptev, T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184 (2000), no. 1, 87–111. [LaW2] A. Laptev, T. Weidl, Recent results on Lieb-Thirring inequalities. Journées ´ “Equations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), Exp. No. XX, Univ. Nantes, Nantes, 2000. odinger equation and the eigenvalue problem. [LiY] P. Li, S.-T. Yau, On the Schr¨ Comm. Math. Phys. 88 (1983), 309–318. [L] E.H. Lieb, The classical limit of quantum spin systems. Comm. Math. Phys. 31 (1973), 327–340. [LT] E.H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schr¨ odinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics, 269–303. Princeton University Press, Princeton, NJ, 1976.

Eigenvalue Estimates for the Aharonov-Bohm Operator [Lu] [MOR]

[P] [ReS] [Ru] [S] [Ti]

137

Y.L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York-LondonToronto, 1962. M. Melgaard, E.-M. Ouhabaz, G. Rozenblum, Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians. Ann. Henri Poincaré 5 (2004), no. 5, 979–1012. Erratum: ibid. 6 (2005), no. 2, 397–398. G. P´ olya, On the eigenvalues of vibrating membranes. Proc. London Math. Soc. 11 (1961), 419–433. M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. 4. Academic Press, New York-London, 1978. S.N.M. Ruijsenaars, The Aharonov-Bohm effect and scattering theory, Ann. Physics 146 (1983), no. 1, 1–34. B. Simon, Trace ideals and their applications, second ed. Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, 2005. E.C. Titchmarsh, Theory of Fourier Integrals, second ed. Oxford University Press, Oxford, 1948.

Rupert L. Frank Department of Mathematics School of Engineering Sciences Royal Institute of Technology 100 44 Stockholm, Sweden e-mail: [email protected] Current address: Department of Mathematics Princeton University Princeton, NJ 08544, USA e-mail: [email protected] Anders M. Hansson Department of Mathematics School of Engineering Sciences Royal Institute of Technology 100 44 Stockholm, Sweden e-mail: [email protected]


The Ablowitz–Ladik Hierarchy Revisited Fritz Gesztesy, Helge Holden, Johanna Michor and Gerald Teschl Abstract. We provide a detailed recursive construction of the Ablowitz–Ladik (AL) hierarchy and its zero-curvature formalism. The two-coefficient AL hierarchy under investigation can be considered a complexified version of the discrete nonlinear Schr¨ odinger equation and its hierarchy of nonlinear evolution equations. Specifically, we discuss in detail the stationary Ablowitz–Ladik formalism in connection with the underlying hyperelliptic curve and the stationary Baker–Akhiezer function and separately the corresponding time-dependent Ablowitz–Ladik formalism. Mathematics Subject Classification (2000). Primary 37K15, 37K10; Secondary 39A12, 35Q55. Keywords. Ablowitz–Ladik hierarchy, discrete NLS.

1. Introduction The prime example of an integrable nonlinear differential-difference system to be discussed in this paper is the Ablowitz–Ladik system, −iαt − (1 − αβ)(α− + α+ ) + 2α = 0, −iβt + (1 − αβ)(β − + β + ) − 2β = 0

(1.1)

with α = α(n, t), β = β(n, t), (n, t) ∈ Z × R. Here we used the notation f ± (n) = f (n ± 1), n ∈ Z, for complex-valued sequences f = {f (n)}n∈Z . The system (1.1) arose in the mid-seventies when Ablowitz and Ladik, in a series of papers [3]–[6] (see also [1], [2, Sect. 3.2.2], [7, Ch. 3], [17]), used inverse scattering methods to analyze certain integrable differential-difference systems. In particular, Ablowitz and Ladik [4] (see also [7, Ch. 3]) showed that in the focusing case, where β = −α, and in the defocusing case, where β = α, (1.1) yields the discrete analog of the Research supported in part by the Research Council of Norway, the US National Science Foundation under Grant No. DMS-0405526, and the Austrian Science Fund (FWF) under Grant No. Y330.

140

F. Gesztesy, H. Holden, J. Michor and G. Teschl

nonlinear Schr¨ odinger equation −iαt − (1 ± |α|2 )(α− + α+ ) + 2α = 0.

(1.2)

Ut + U V − V + U = 0.

(1.4)

We will refer to (1.1) as the Ablowitz–Ladik system. The principal theme of this paper will be to derive a detailed recursive construction of the Ablowitz–Ladik hierarchy, a completely integrable sequence of systems of nonlinear evolution equations on the lattice Z whose first nonlinear member is the Ablowitz–Ladik system (1.1). In addition, we discuss the zero-curvature formalism for the Ablowitz–Ladik (AL) hierarchy in detail. Since the original discovery of Ablowitz and Ladik in the mid-seventies, there has been great interest in the area of integrable differential-difference equations. Two principal directions of research are responsible for this development: Originally, the development was driven by the theory of completely integrable systems and its applications to fields such as nonlinear optics, and more recently, it gained additional momentum due to its intimate connections with the theory of orthogonal polynomials. In this paper we will not discuss the connection with orthogonal polynomials (see, however, the introduction of [31]) and instead refer to the recent references [13], [20], [37], [38], [42], [43], [44], [47], [48], [49], and the literature cited therein. The first systematic discussion of the Ablowitz–Ladik (AL) hierarchy appears to be due to Schilling [45] (cf. also [51], [55], [58]); infinitely many conservation laws are derived, for instance, by Ding, Sun, and Xu [21]; the bi-Hamiltonian structure of the AL hierarchy is considered by Ercolani and Lozano [23]; connections between the AL hierarchy and the motion of a piecewise linear curve have been established by Doliwa and Santini [22]; B¨ acklund and Darboux transformations were studied by Geng [26] and Vekslerchik [56]; the Hirota bilinear formalism, AL τ -functions, etc., were considered by Vekslerchik [55]. The initial value problem for half-infinite AL systems was discussed by Common [19], for an application of the inverse scattering method to (1.2) we refer to Vekslerchik and Konotop [57]. This just scratches the surface of these developments and the interested reader will find much more material in the references cited in these papers and the ones discussed below. Algebro-geometric (and periodic) solutions of the AL system (1.1) have briefly been studied by Ahmad and Chowdhury [8], [9], Bogolyubov, Prikarpatskii, and Samoilenko [14], Bogolyubov and Prikarpatskii [15], Chow, Conte, and Xu [18], Geng, Dai, and Cao [27], and Vaninsky [53]. In an effort to analyze models describing oscillations in nonlinear dispersive wave systems, Miller, Ercolani, Krichever, and Levermore [40] (see also [39]) gave a detailed analysis of algebro-geometric solutions of the AL system (1.1). Introducing

α − α− z −1 z α z − 1 − αβ − (1.3) U (z) = , V (z) = i β−z − β 1 + α− β − z −1 βz 1 with z ∈ C \ {0} a spectral parameter, the authors in [40] relied on the fact that the Ablowitz–Ladik system (1.1) is equivalent to the zero-curvature equations

The Ablowitz–Ladik Hierarchy Revisited

141

Miller, Ercolani, Krichever, and Levermore [40] then derived the theta function representations of α, β satisfying the AL system (1.1). Vekslerchik [54] also studied finite-genus solutions for the AL hierarchy by establishing connections with Fay’s identity for theta functions. Recently, a detailed study of algebro-geometric solutions for the entire AL hierarchy has been provided in [31]. The latter reference also contains an extensive discussion of the connection between the Ablowitz–Ladik system (1.1) and orthogonal polynomials on the unit circle. The algebro-geometric initial value problem for the Ablowitz–Ladik hierarchy with complex-valued initial data, that is, the construction of α and β by starting from a set of initial data (nonspecial divisors) of full measure, will be presented in [32]. The Hamiltonian and Lax formalisms for the AL hierarchy will be revisited in [33]. In addition to these recent developments on the AL system and the AL hierarchy, we offer a variety of results in this paper apparently not covered before. These include: • An effective recursive construction of the AL hierarchy using Laurent polynomials. • The detailed connection between the AL hierarchy and a “complexified” version of transfer matrices first introduced by Baxter [11], [12]. • A detailed treatment of the stationary and time-dependent Ablowitz–Ladik formalism. The structure of this paper is as follows: In Section 2 we describe our zerocurvature formalism for the Ablowitz–Ladik (AL) hierarchy. Extending a recursive polynomial approach discussed in great detail in [29] in the continuous case and in [16], [30, Ch. 4], [52, Chs. 6, 12] in the discrete context to the case of Laurent polynomials with respect to the spectral parameter, we derive the AL hierarchy of systems of nonlinear evolution equations whose first nonlinear member is the Ablowitz–Ladik system (1.1). Section 3 is devoted to a detailed study of the stationary AL hierarchy. We employ the recursive Laurent polynomial formalism of Section 2 to describe nonnegative divisors of degree p on a hyperelliptic curve Kp of genus p associated with the pth system in the stationary AL hierarchy. The corresponding time-dependent results for the AL hierarchy are presented in detail in Section 4. Finally, Appendix A is of a technical nature and summarizes expansions of various key quantities related to the Laurent polynomial recursion formalism as the spectral parameter tends to zero or to infinity.

2. The Ablowitz–Ladik hierarchy, recursion relations, zero-curvature pairs, and hyperelliptic curves In this section we provide the construction of the Ablowitz–Ladik hierarchy employing a polynomial recursion formalism and derive the associated sequence of Ablowitz–Ladik zero-curvature pairs. Moreover, we discuss the hyperelliptic curve underlying the stationary Ablowitz–Ladik hierarchy. We denote by CZ the set of complex-valued sequences indexed by Z.

142

F. Gesztesy, H. Holden, J. Michor and G. Teschl Throughout this section we suppose the following hypothesis.

Hypothesis 2.1. In the stationary case we assume that α, β satisfy α, β ∈ CZ ,

α(n)β(n) ∈ / {0, 1}, n ∈ Z.

(2.1)

In the time-dependent case we assume that α, β satisfy α( · , t), β( · , t) ∈ CZ , t ∈ R,

α(n, · ), β(n, · ) ∈ C 1 (R), n ∈ Z,

α(n, t)β(n, t) ∈ / {0, 1}, (n, t) ∈ Z × R.

(2.2)

Actually, up to Remark 2.11 our analysis will be time-independent and hence only the lattice variations of α and β will matter. We denote by S ± the shift operators acting on complex-valued sequences f = {f (n)}n∈Z ∈ CZ according to (S ± f )(n) = f (n ± 1),

n ∈ Z.

(2.3)

Moreover, we will frequently use the notation f ± = S ± f,

f ∈ CZ .

(2.4)

To construct the Ablowitz–Ladik hierarchy we will try to generalize (1.3) by considering the 2 × 2 matrix

z α U (z) = , z ∈ C, (2.5) zβ 1 and making the ansatz Vp (z) = i

0

G− p (z)

−Fp− (z)

1

Hp− (z) −Kp− (z)

,

p = (p− , p+ ) ∈ N20 ,

(2.6)

where Gp , Kp , Fp , and Hp are chosen as Laurent polynomials1 (suggested by the appearance of z −1 in the matrix V in (1.3)) Gp (z) =

p− ℓ=1 p−

Fp (z) =

ℓ=1

Hp (z) =

p− ℓ=1

Kp (z) =

p− ℓ=1

z −ℓ gp− −ℓ,− + z −ℓ fp− −ℓ,− + z

−ℓ

hp− −ℓ,− +

z −ℓ kp− −ℓ,− +

p+ ℓ=0 p+

ℓ=0 p+

ℓ=0

p+ ℓ=0

z ℓ gp+ −ℓ,+ , z ℓ fp+ −ℓ,+ , (2.7) z ℓ hp+ −ℓ,+ , z ℓ kp+ −ℓ,+ .

1 In this paper, a sum is interpreted as zero whenever the upper limit in the sum is strictly less than its lower limit.


143

Without loss of generality we will only look at the time-independent case and add time later on. Then the stationary zero-curvature equation, 0 = U Vp − Vp+ U,

(2.8)

is equivalent to the following relationships between the Laurent polynomials 1 0 − Fp − zFp− − α(Gp + Kp− ) z(G− p − Gp ) + zβFp + αHp , U Vp − Vp+ U = i − zβ(G− −zβFp− − αHp + Kp − Kp− p + Kp ) − zHp + Hp (2.9) respectively, to − z(G− p − Gp ) + zβFp + αHp = 0,

zβFp−

+ αHp − Kp +

Kp−

(2.10)

= 0,

(2.11)

−Fp + zFp− + α(Gp + Kp− ) = 0,

(2.12)

− zβ(G− p + Kp ) − zHp + Hp = 0.

(2.13)

Lemma 2.2. Suppose the Laurent polynomials defined in (2.7) satisfy the zerocurvature equation (2.8), then f0,+ = 0, kℓ,± −

− kℓ,±

h0,− = 0, = gℓ,± −

− g0,± = g0,± ,

− gℓ,± ,

− k0,± = k0,± ,

ℓ = 0, . . . , p± − 1,

(2.14)

gp+ ,+ − gp−+ ,+ = kp+ ,+ − kp−+ ,+ . (2.15)

Proof. Comparing coefficients at the highest order of z in (2.11) and the lowest in − − (2.10) immediately yields f0,+ = 0, h0,− = 0. Then g0,+ = g0,+ , k0,− = k0,− are necessarily lattice constants by (2.10), (2.11). Since det(U (z)) = 0 for z ∈ C \ {0} by (2.1), (2.8) yields tr(Vp+ ) = tr(U Vp U −1 ) = tr(Vp ) and hence − Gp − G− p = Kp − Kp ,

implying (2.15). Taking ℓ = 0 in (2.15) then yields g0,− =

(2.16) − g0,−

and k0,+ =

− k0,+ .

In particular, this lemma shows that we can choose kℓ,± = gℓ,± , 0 ≤ ℓ ≤ p± − 1,

kp+ ,+ = gp+ ,+

(2.17)

without loss of generality (since this can always be achieved by adding a Laurent polynomial times the identity to Vp , which does not affect the zero-curvature equation). Hence the ansatz (2.7) can be refined as follows (it is more convenient in the following to re-label hp+ ,+ = hp− −1,− and kp+ ,+ = gp− ,− , and hence, gp− ,− = gp+ ,+ ), Fp (z) =

p− ℓ=1 p−

Gp (z) =

ℓ=1

fp− −ℓ,− z −ℓ + gp− −ℓ,− z −ℓ +

p+ −1

ℓ=0 p+

ℓ=0

fp+ −1−ℓ,+ z ℓ ,

gp+ −ℓ,+ z ℓ ,

(2.18) (2.19)

144


Hp (z) =

p− −1

ℓ=0

hp− −1−ℓ,− z −ℓ +

p+ ℓ=1

hp+ −ℓ,+ z ℓ ,

Kp (z) = Gp (z) since gp− ,− = gp+ ,+ .

(2.20) (2.21)

In particular, (2.21) renders Vp in (2.6) traceless in the stationary context. We emphasize, however, that equation (2.21) ceases to be valid in the time-dependent context: In the latter case (2.21) needs to be replaced by Kp (z) =

p− ℓ=0

gp− −ℓ,− z −ℓ +

p+ ℓ=1

gp+ −ℓ,+ z ℓ = Gp (z) + gp− ,− − gp+ ,+ .

(2.22)

Plugging the refined ansatz (2.18)–(2.21) into the zero-curvature equation (2.8) and comparing coefficients then yields the following result. Lemma 2.3. Suppose that U and Vp satisfy the zero-curvature equation (2.8). Then the coefficients {fℓ,± }ℓ=0,...,p± −1 , {gℓ,± }ℓ=0,...,p± , and {hℓ,± }ℓ=0,...,p± −1 of Fp , Gp , Hp , and Kp in (2.18)–(2.21) satisfy the following relations g0,+ = 12 c0,+ , gℓ+1,+ −

− gℓ+1,+ − fℓ+1,+

=

αh− ℓ,+

f0,+ = −c0,+ α+ ,

+ βfℓ,+ ,

h0,+ = c0,+ β,

0 ≤ ℓ ≤ p+ − 1,

− = fℓ,+ − α(gℓ+1,+ + gℓ+1,+ ),

− hℓ+1,+ = h− ℓ,+ + β(gℓ+1,+ + gℓ+1,+ ),

0 ≤ ℓ ≤ p+ − 2,

0 ≤ ℓ ≤ p+ − 2,

(2.23) (2.24) (2.25) (2.26)

and g0,− = 12 c0,− ,

f0,− = c0,− α,

− − gℓ+1,− − gℓ+1,− = αhℓ,− + βfℓ,− ,

h0,− = −c0,− β + ,

0 ≤ ℓ ≤ p− − 1,

− − fℓ+1,− = fℓ,− + α(gℓ+1,− + gℓ+1,− ),

− h− ℓ+1,− = hℓ,− − β(gℓ+1,− + gℓ+1,− ),

0 ≤ ℓ ≤ p− − 2,

0 ≤ ℓ ≤ p− − 2.

(2.27) (2.28) (2.29) (2.30)

Here c0,± ∈ C are given constants. In addition, (2.8) reads 0 = U Vp − Vp+ U ⎛

⎞ −α(gp+ ,+ + gp−− ,− ) 0 − ⎜ +fp+ −1,+ − fp− −1,− ⎟ ⎜ ⎟ = i⎜ ⎟. − ⎝ z(β(gp+ ,+ + gp− ,− ) ⎠ 0 −hp− −1,− + h− p+ −1,+ )

(2.31)

Given Lemma 2.3, we now introduce the sequences {fℓ,± }ℓ∈N0 , {gℓ,± }ℓ∈N0 , and {hℓ,± }ℓ∈N0 recursively by g0,+ = 12 c0,+ ,

− gℓ+1,+ − gℓ+1,+ − fℓ+1,+

=

αh− ℓ,+

f0,+ = −c0,+ α+ ,

+ βfℓ,+ ,

ℓ ∈ N0 ,

= fℓ,+ − α(gℓ+1,+ +

− gℓ+1,+ ),

h0,+ = c0,+ β,

(2.32) (2.33)

ℓ ∈ N0 ,

(2.34)

The Ablowitz–Ladik Hierarchy Revisited − hℓ+1,+ = h− ℓ,+ + β(gℓ+1,+ + gℓ+1,+ ),

ℓ ∈ N0 ,

145 (2.35)

and g0,− = 12 c0,− , gℓ+1,− −

− gℓ+1,−

= αhℓ,− +

f0,− = c0,− α, − βfℓ,− ,

ℓ ∈ N0 ,

h0,− = −c0,− β + ,

− − fℓ+1,− = fℓ,− + α(gℓ+1,− + gℓ+1,− ),

h− ℓ+1,−

= hℓ,− − β(gℓ+1,− +

− gℓ+1,− ),

ℓ ∈ N0 ,

ℓ ∈ N0 .

(2.36) (2.37) (2.38) (2.39)

For later use we also introduce f−1,± = h−1,± = 0.

(2.40)

Remark 2.4. The sequences {fℓ,+ }ℓ∈N0 , {gℓ,+ }ℓ∈N0 , and {hℓ,+ }ℓ∈N0 can be computed recursively as follows: Assume that fℓ,+ , gℓ,+ , and hℓ,+ are known. Equation (2.33) is a first-order difference equation in gℓ+1,+ that can be solved directly and yields a local lattice function that is determined up to a new constant denoted by cℓ+1,+ ∈ C. Relations (2.34) and (2.35) then determine fℓ+1,+ and hℓ+1,+ , etc. The sequences {fℓ,− }ℓ∈N0 , {gℓ,− }ℓ∈N0 , and {hℓ,− }ℓ∈N0 are determined similarly. Upon setting one explicitly obtains

γ = 1 − αβ,

(2.41)

f0,+ = c0,+ (−α+ ), f1,+ = c0,+ − γ + α++ + (α+ )2 β + c1,+ (−α+ ), g0,+ = 12 c0,+ ,

g1,+ = c0,+ (−α+ β) + 12 c1,+ , g2,+ = c0,+ (α+ β)2 − γ + α++ β − γα+ β − + c1,+ (−α+ β) + 12 c2,+ ,

h0,+ = c0,+ β, h1,+ = c0,+ γβ − − α+ β 2 + c1,+ β, f0,− = c0,− α, f1,− = c0,− γα− − α2 β + + c1,− α,

(2.42)

g0,− = 21 c0,− ,

g1,− = c0,− (−αβ + ) + 12 c1,− , g2,− = c0,− (αβ + )2 − γ + αβ ++ − γα− β + + c1,− (−αβ + ) + 21 c2,− ,

h0,− = c0,− (−β + ), h1,− = c0,− − γ + β ++ + α(β + )2 + c1,− (−β + ), etc.

Here {cℓ,± }ℓ∈N denote summation constants which naturally arise when solving the difference equations for gℓ,± in (2.33), (2.37).

146


In particular, by (2.31), the stationary zero-curvature relation (2.8), 0 = U Vp − Vp+ U , is equivalent to −α(gp+ ,+ + gp−− ,− ) + fp+ −1,+ − fp−− −1,− = 0,

(2.43)

β(gp−+ ,+ + gp− ,− ) + h− p+ −1,+ − hp− −1,− = 0.

(2.44)

Thus, varying p± ∈ N0 , equations (2.43) and (2.44) give rise to the stationary Ablowitz–Ladik (AL) hierarchy which we introduce as follows

−α(gp+ ,+ + gp−− ,− ) + fp+ −1,+ − fp−− −1,− = 0, s-ALp (α, β) = − − β(gp+ ,+ + gp− ,− ) + hp+ −1,+ − hp− −1,− (2.45) p = (p− , p+ ) ∈ N20 . Explicitly (recalling γ = 1 − αβ and taking p− = p+ for simplicity),

−c(0,0) α = 0, s-AL(0,0) (α, β) = c(0,0) β

−γ(c0,− α− + c0,+ α+ ) − c(1,1) α = 0, s-AL(1,1) (α, β) = − + γ(c0,+ β + c0,− β ) + c(1,1) β ⎛ ⎞ ++ + −− − −γ c0,+ α γ + c0,− α γ − α(c0,+ α+ β− + c0,− α− β + ) − 2 + 2 ⎜ ⎟ (α ) + c (α ) ) −β(c 0,− 0,+ ⎟ s-AL(2,2) (α, β) = ⎜ ⎝ γ c0,− β ++ γ + + c0,+ β −− γ − − β(c0,+ α+ β − + c0,− α− β + ) ⎠ −α(c0,+ (β − )2 + c0,− (β + )2 )

− −γ(c1,− α + c1,+ α+ ) − c(2,2) α = 0, etc., (2.46) + − + γ(c1,+ β + c1,− β ) + c(2,2) β represent the first few equations of the stationary Ablowitz–Ladik hierarchy. Here we introduced cp = (cp,− + cp,+ )/2, p± ∈ N0 . (2.47) By definition, the set of solutions of (2.45), with p± ranging in N0 and cℓ,± ∈ C, ℓ ∈ N0 , represents the class of algebro-geometric Ablowitz–Ladik solutions. In the special case p = (1, 1), c0,± = 1, and c(1,1) = −2, one obtains the stationary version of the Ablowitz–Ladik system (1.1)

−γ(α− + α+ ) + 2α = 0. (2.48) − + γ(β + β ) − 2β

Subsequently, it will also be useful to work with the corresponding homogeˆ ℓ,± , defined by the vanishing of all summation neous coefficients fˆℓ,± , gˆℓ,± , and h constants ck,± for k = 1, . . . , ℓ, and choosing c0,± = 1, fˆ0,+ = −α+ , gˆ0,± = 21 ,

ˆ 0,+ = β, h

fˆ0,− = α,

fˆℓ,± = fℓ,± |c0,± =1, cj,± =0,j=1,...,ℓ ,

gˆℓ,± = gℓ,± |c0,± =1, cj,± =0,j=1,...,ℓ , ˆ 0,− = −β + , h

ℓ ∈ N,

ˆ ℓ,± = hℓ,± |c0,± =1, cj,± =0,j=1,...,ℓ , h

ℓ ∈ N, ℓ ∈ N.

(2.49) (2.50) (2.51)


147

By induction one infers that fℓ,± =

ℓ

k=0

cℓ−k,± fˆk,± ,

gℓ,± =

ℓ

cℓ−k,± gˆk,± ,

hℓ,± =

k=0

ℓ

ˆ k,± . (2.52) cℓ−k,± h

k=0

In a slight abuse of notation we will occasionally stress the dependence of fℓ,± , gℓ,± , and hℓ,± on α, β by writing fℓ,± (α, β), gℓ,± (α, β), and hℓ,± (α, β). Remark 2.5. Using the nonlinear recursion relations (A.29)–(A.34) recorded in Theorem A.1, one infers inductively that all homogeneous elements fˆℓ,± , gˆℓ,± , and ˆ ℓ,± , ℓ ∈ N0 , are polynomials in α, β, and some of their shifts. (Alternatively, h one can prove directly by induction that the nonlinear recursion relations (A.29)– (A.34) are equivalent to that in (2.32)–(2.39) with all summation constants put equal to zero, cℓ,± = 0, ℓ ∈ N.) Remark 2.6. As an efficient tool to later distinguish between nonhomogeneous and ˆ ℓ,± , respectively, we now homogeneous quantities fℓ,± , gℓ,± , hℓ,± , and fˆℓ,± , gˆℓ,± , h introduce the notion of degree as follows. Denote / r ≥ 0, (S + )r , r ∈ Z, (2.53) f (r) = S (r) f, f = {f (n)}n∈Z ∈ CZ , S (r) = (S − )−r , r < 0, and define

deg α(r) = r,

deg β (r) = −r,

This then results in (r) (r) deg fˆℓ,+ = ℓ + 1 + r, deg fˆℓ,− = −ℓ + r, (r) (r) ˆ ˆ deg h deg h ℓ,+ = ℓ − r, ℓ,− = −ℓ − 1 − r,

r ∈ Z. (r) deg gˆℓ,± = ±ℓ, ℓ ∈ N0 , r ∈ Z,

(2.54)

(2.55)

using induction in the linear recursion relations (2.32)–(2.39).

In accordance with our notation introduced in (2.49)–(2.51), the corresponding homogeneous stationary Ablowitz–Ladik equations are defined by < p (α, β) = s-ALp (α, β) s-AL , p = (p− , p+ ) ∈ N20 . (2.56) c0,± =1, c =0, ℓ=1,...,p± ℓ,±

We also note the following useful result.

Lemma 2.7. The coefficients fℓ,± , gℓ,± , and hℓ,± satisfy the relations − − gℓ,+ − gℓ,+ = αhℓ,+ + βfℓ,+ ,

− gℓ,− − gℓ,− = αh− ℓ,− + βfℓ,− ,

ℓ ∈ N0 ,

ℓ ∈ N0 .

(2.57)

Moreover, we record the following symmetries, ˆ ℓ,∓ (c0,∓ , β, α), fˆℓ,± (c0,± , α, β) = h

gˆℓ,± (c0,± , α, β) = gˆℓ,∓ (c0,∓ , β, α),

ℓ ∈ N0 . (2.58)

148


Proof. The relations (2.57) are derived as follows: − − − αhℓ+1,+ + βfℓ+1,+ = αh− ℓ,+ + αβ(gℓ+1,+ + gℓ+1,+ ) + βfℓ,+ − αβ(gℓ+1,+ + gℓ+1,+ )

= αh− ℓ,+ + βfℓ,+

− , = gℓ+1,+ − gℓ+1,+

(2.59)

and − − − αh− ℓ+1,− + βfℓ+1,− = αhℓ,− − αβ(gℓ+1,− + gℓ+1,− ) + βfℓ,− + αβ(gℓ+1,− + gℓ+1,− ) − = αhℓ,− + βfℓ,−

− = gℓ+1,− + gℓ+1,− .

(2.60)

ˆ ℓ,∓ (β, α) and gˆℓ,∓ (β, α) satisfy The statement (2.58) follows by showing that h the same recursion relations as those of fˆℓ,± (α, β) and gˆℓ,± (α, β), respectively. That the recursion constants are the same, follows from the observation that the corresponding coefficients have the proper degree. Next we turn to the Laurent polynomials Fp , Gp , Hp , and Kp defined in (2.18)–(2.20) and (2.22). Explicitly, one obtains F(0,0) = 0, F(1,1) = c0,− αz −1 + c0,+ (−α+ ), F(2,2) = c0,− αz −2 + c0,− γα− − α2 β + + c1,− α z −1 + c0,+ − γ + α++ + (α+ )2 β + c1,+ (−α+ ) + c0,+ (−α+ )z,

G(0,0) = 12 c0,+ ,

G(1,1) = 12 c0,− z −1 + c0,+ (−α+ β) + 12 c1,+ + 12 c0,+ z, G(2,2) = 12 c0,− z −2 + c0,− (−αβ + ) + 21 c1,− z −1 + 2 + c0,+ (α β) − γ + α++ β − γα+ β − + c1,+ (−α+ β) + 12 c2,+ + c0,+ (−α+ β) + 21 c1,+ z + 21 c0,+ z 2 ,

H(0,0) = 0,

H(1,1) = c0,− (−β + ) + c0,+ βz,

H(2,2) = c0,− (−β + )z −1 + c0,− − γ + β ++ + α(β + )2 + c1,− (−β + ) + c0,+ (γβ − − α+ β 2 ) + c1,+ β z + c0,+ βz 2 ,

K(0,0) = 21 c0,− ,

K(1,1) = 21 c0,− z −1 + c0,− (−αβ + ) + 12 c1,− + 21 c0,+ z, K(2,2) = 21 c0,− z −2 + c0,− (−αβ + ) + 21 c1,− z −1 + c0,− (αβ + )2 − γ + αβ ++ − γα− β + + c1,− (−αβ + ) + 21 c2,− + c0,+ (−α+ β) + 12 c1,+ z + 21 c0,+ z 2 , etc.

(2.61)


149

The corresponding homogeneous quantities are defined by (ℓ ∈ N0 ) F*0,∓ (z) = 0,

* 0,− (z) = 0, G

*0,+ (z) = 1 , G 2

* 0,∓ (z) = 0, H

* 0,− (z) = 1 , K 2 * 0,+ (z) = 0, K

F*ℓ,− (z) =

* ℓ,− (z) = G

ℓ

k=1 ℓ

* ℓ,− (z) = K

* ℓ,+ (z) = K

gˆℓ−k,− z −k ,

F*ℓ,+ (z) =

ℓ−1

fˆℓ−1−k,+ z k ,

k=0

k=1

* ℓ,+ (z) = G

* ℓ,− (z) = H

fˆℓ−k,− z −k ,

ℓ

gˆℓ−k,+ z k ,

k=0 ℓ−1

ˆ ℓ−1−k,− z −k , h

k=0 ℓ

k=0 ℓ

k=1

* ℓ,+ (z) = H

ℓ

(2.62) ˆ ℓ−k,+ z k , h

k=1

*ℓ,− (z) + gˆℓ,− , gˆℓ−k,− z −k = G

*ℓ,+ (z) − gˆℓ,+ . gˆℓ−k,+ z k = G

Similarly, with Fℓ+ ,+ , Gℓ+ ,+ , Hℓ+ ,+ , and Kℓ+ ,+ denoting the polynomial parts of Fℓ , Gℓ , Hℓ , and Kℓ , respectively, and Fℓ− ,− , Gℓ− ,− , Hℓ− ,− , and Kℓ− ,− denoting the Laurent parts of Fℓ , Gℓ , Hℓ , and Kℓ , ℓ = (ℓ− , ℓ+ ) ∈ N0 , such that Fℓ (z) = Fℓ− ,− (z) + Fℓ+ ,+ (z),

Gℓ (z) = Gℓ− ,− (z) + Gℓ+ ,+ (z),

Hℓ (z) = Hℓ− ,− (z) + Hℓ+ ,+ (z),

Kℓ (z) = Kℓ− ,− (z) + Kℓ+ ,+ (z),

(2.63)

one finds that Fℓ± ,± =

ℓ±

k=1

Gℓ− ,− =

ℓ−

k=1

Kℓ− ,− =

ℓ−

k=0

cℓ± −k,± F*k,± ,

* k,− , cℓ− −k,− G

* k,− , cℓ− −k,− K

Hℓ± ,± =

ℓ±

k=1

Gℓ+ ,+ =

ℓ+

* k,± , cℓ± −k,± H

k=0

Kℓ+ ,+ =

ℓ+

k=1

*k,+ , cℓ+ −k,+ G

(2.64)

* k,+ . cℓ+ −k,+ K

In addition, one immediately obtains the following relations from (2.58): Lemma 2.8. Let ℓ ∈ N0 . Then, * ℓ,∓ (β, α, z −1 , n), F*ℓ,± (α, β, z, n) = H * ℓ,± (α, β, z, n) = F*ℓ,∓ (β, α, z −1 , n), H

* ℓ,∓ (β, α, z −1 , n), * ℓ,± (α, β, z, n) = G G

* ℓ,∓ (β, α, z −1 , n). * ℓ,± (α, β, z, n) = K K

(2.65) (2.66) (2.67) (2.68)

150


Returning to the stationary Ablowitz–Ladik hierarchy, we will frequently assume in the following that α, β satisfy the pth stationary Ablowitz–Ladik system s-ALp (α, β) = 0, supposing a particular choice of summation constants cℓ,± ∈ C, ℓ = 0, . . . , p± , p± ∈ N0 , has been made. Remark 2.9. (i) The particular choice c0,+ = c0,− = 1 in (2.45) yields the stationary Ablowitz– Ladik equation. Scaling c0,± with the same constant then amounts to scaling Vp with this constant which drops out in the stationary zero-curvature equation (2.8). (ii) Different ratios between c0,+ and c0,− will lead to different stationary hierarchies. In particular, the choice c0,+ = 2, c0,− = · · · = cp− −1,− = 0, cp− ,− = 0, yields the stationary Baxter–Szeg˝o hierarchy considered in detail in [28]. However, in this case some parts from the recursion relation for the negative coefficients still remain. In fact, (2.39) reduces to gp− ,− − gp−− ,− = αhp− −1,− , hp− −1,− = 0 and thus requires gp− ,− to be a constant in (2.45) and (2.87). Moreover, fp− −1,− = 0 in (2.45) in this case. (iii) Finally, by Lemma 2.8, the choice c0,+ = · · · = cp+ −1,+ = 0, cp+ ,+ = 0, c0,− = 2 again yields the Baxter–Szeg˝o hierarchy, but with α and β interchanged. Next, taking into account (2.21), one infers that the expression Rp , defined as Rp = G2p − Fp Hp ,

(2.69)

is a lattice constant, that is, Rp − Rp− = 0, since taking determinants in the stationary zero-curvature equation (2.8) immediately yields 2 − − 2 γ − (G− (2.70) p ) + Fp Hp + Gp − Fp Hp z = 0. Hence, Rp (z) only depends on z, and assuming in addition to (2.1) that c0,± ∈ C \ {0},

p = (p− , p+ ) ∈ N20 \ {(0, 0)},

(2.71)

one may write Rp as2 Rp (z) =

c0,+ 2z p−

2 2p+1 (z − Em ), m=0

{Em }2p+1 m=0 ⊂ C \ {0}, p = p− + p+ − 1 ∈ N0 . (2.72)

Moreover, (2.69) also implies lim 4z 2p− Rp (z) = c20,+

z→0

2p+1

(−Em ) = c20,− ,

(2.73)

m=0

2 We use the convention that a product is to be interpreted equal to 1 whenever the upper limit of the product is strictly less than its lower limit.


151

and hence, 2p+1

Em =

m=0

c20,− . c20,+

(2.74)

Relation (2.69) allows one to introduce a hyperelliptic curve Kp of (arithmetic) genus p = p− + p+ − 1 (possibly with a singular affine part), where 2p− Kp : Fp (z, y) = y 2 − 4c−2 Rp (z) 0,+ z

= y2 −

2p+1 m=0

(z − Em ) = 0,

p = p− + p+ − 1.

(2.75)

Remark 2.10. In the special case p− = p+ and cℓ,+ = cℓ,− , ℓ = 0, . . . , p− , the symmetries of Lemma 2.8 also hold for Fp , Gp , and Hp and thus Rp (1/z) = Rp (z) and hence the numbers Em , m = 0, . . . , 2p + 1, come in pairs (Ek , 1/Ek ), k = 1, . . . , p + 1. Equations (2.10)–(2.13) and (2.69) permit one to derive nonlinear difference equations for Fp , Gp , and Hp separately. One obtains

(α+ + zα)2 Fp − z(α+ )2 γFp−

2

− 2zα2 γ + (α+ + zα)2 Fp + z(α+ )2 γFp− Fp+

+ z 2 α4 (γ + )2 (Fp+ )2 = 4(αα+ )2 (α+ + αz)2 Rp , +

+

+

(α + zα)(β + zβ )(z + α β)(1 + zαβ

+

(2.76)

)G2p

+ + + − + + + z(α+ γG− p + zαγ Gp )(zβ γGp + βγ Gp ) + 2 + + − zγ (α β + z αβ )(2 − γ ) + 2z(1 − γ + )(2 − γ) G− p Gp + + + + 2 + − zγ 2z(1 − γ)(2 − γ ) + (α β + z αβ )(2 − γ) Gp Gp

= (α+ β − z 2 αβ + )2 Rp , (2.77) 2 z 2 (β + )2 γHp− − β 2 γ + Hp+ − 2z(β + zβ + )2 (β + )2 γHp− + β 2 γ + Hp+ Hp + (β + zβ + )4 Hp2 = 4z 2 (ββ + )2 (β + β + z)2 Rp .

(2.78)

Equations analogous to (2.76)–(2.78) can be used to derive nonlinear recurˆ ℓ,± (i.e., the ones sion relations for the homogeneous coefficients fˆℓ,± , gˆℓ,± , and h satisfying (2.49)–(2.51) in the case of vanishing summation constants) as proved ˆ ℓ,± in Theorem A.1 in Appendix A. This then yields a proof that fˆℓ,± , gˆℓ,± , and h are polynomials in α, β, and some of their shifts (cf. Remark 2.5). In addition, as proven in Theorem A.2, (2.76) leads to an explicit determination of the summation constants c1,± , . . . , cp± ,± in (2.45) in terms of the zeros E0 , . . . , E2p+1 of the associated Laurent polynomial Rp in (2.72). In fact, one can prove (cf. (A.42)) cℓ,± = c0,± cℓ E ±1 , ℓ = 0, . . . , p± , (2.79)

152


where c0 E ±1 = 1, ±1 ck E =−

k

(2.80) (2j0 )! · · · (2j2p+1 )! ±j2p+1 E ±j0 · · · E2p+1 , 22k (j0 !)2 · · · (j2p+1 !)2 (2j0 − 1) · · · (2j2p+1 − 1) 0

j0 ,...,j2p+1 =0 j0 +···+j2p+1 =k

k ∈ N,

±1 are symmetric functions of E ±1 = (E0±1 , . . . , E2p+1 ) introduced in (A.5) and (A.6).

Remark 2.11. If α, β satisfy one of the stationary Ablowitz–Ladik equations in (2.45) for a particular value of p, s-ALp (α, β) = 0, then they satisfy infinitely many such equations of order higher than p for certain choices of summation constants cℓ,± . This can be shown as in [29, Remark I.1.5]. Finally we turn to the time-dependent Ablowitz–Ladik hierarchy. For that purpose the coefficients α and β are now considered as functions of both the lattice point and time. For each system in the hierarchy, that is, for each p, we introduce a deformation (time) parameter tp ∈ R in α, β, replacing α(n), β(n) by α(n, tp ), β(n, tp ). Moreover, the definitions (2.5), (2.6), and (2.18)–(2.20) of U, Vp , and Fp , Gp , Hp , Kp , respectively, still apply; however, equation (2.21) now needs to be replaced by (2.22) in the time-dependent context. Imposing the zero-curvature relation Utp + U Vp − Vp+ U = 0,

p ∈ N20 ,

(2.81)

then results in the equations 0 = Utp + U Vp − Vp+ U 1 0 − −iαtp + Fp − zFp− − α(Gp + Kp− ) z(G− p − Gp ) + zβFp + αHp =i − −zβFp− − αHp + Kp − Kp− −izβtp + zβ(G− p + Kp ) − zHp + Hp ⎛ ⎞ −iαtp − α(gp+ ,+ + gp−− ,− ) 0 − ⎟ ⎜ − f +f p+ −1,+ p− −1,− ⎟, (2.82) = i⎜ ⎠ ⎝z − iβtp + β(gp− ,+ + gp− ,− ) + 0 −hp− −1,− + h− p+ −1,+ or equivalently,

αtp = i zFp− + α(Gp + Kp− ) − Fp , −1 − Hp , βtp = −i β(G− p + K p ) − Hp + z 0=

z(G− p

0=

zβFp−

− Gp ) + zβFp + + αHp +

Kp−

αHp− ,

− Kp .

(2.83) (2.84) (2.85) (2.86)

The Ablowitz–Ladik Hierarchy Revisited Varying p ∈ N20 , the collection of evolution equations

−iαtp − α(gp+ ,+ + gp−− ,− ) + fp+ −1,+ − fp−− −1,− = 0, ALp (α, β) = − − −iβtp + β(gp+ ,+ + gp− ,− ) − hp− −1,− + hp+ −1,+

153

(2.87)

tp ∈ R, p = (p− , p+ ) ∈ N20 ,

then defines the time-dependent Ablowitz–Ladik hierarchy. Explicitly, taking p− = p+ for simplicity,

−iαt(0,0) − c(0,0) α = 0, AL(0,0) (α, β) = −iβt(0,0) + c(0,0) β

−iαt(1,1) − γ(c0,− α− + c0,+ α+ ) − c(1,1) α = 0, AL(1,1) (α, β) = − + −iβt(1,1) + γ(c0,+ β + c0,− β ) + c(1,1) β

AL(2,2) (α, β) (2.88) ⎛ ⎞ −iαt(2,2) − γ c0,+ α++ γ + + c0,− α−− γ − − α(c0,+α+ β − + c0,− α− β + ) − 2 + 2 ⎜ ⎟ −β(c0,− (α ) + c0,+ (α ) ) ⎟ =⎜ ++ + −− − + − − + ⎠ ⎝ −iβt + γ c0,− β γ + c0,+ β γ − β(c0,+α β + c0,− α β ) (2,2) − 2 + 2 −α(c0,+ (β ) + c0,− (β ) )

−γ(c1,− α− + c1,+ α+ ) − c(2,2) α = 0, etc., + γ(c1,+ β − + c1,− β + ) + c(2,2) β

represent the first few equations of the time-dependent Ablowitz–Ladik hierarchy. Here we recall the definition of cp in (2.47). The special case p = (1, 1), c0,± = 1, and c(1,1) = −2, that is,

−iαt(1,1) − γ(α− + α+ ) + 2α = 0, (2.89) − + −iβt(1,1) + γ(β + β ) − 2β represents the Ablowitz–Ladik system (1.1). The corresponding homogeneous equations are then defined by < ALp (α, β) = ALp (α, β)c0,± =1, c =0, ℓ=1,...,p± = 0, p = (p− , p+ ) ∈ N20 . (2.90) ℓ,±

By (2.87), (2.33), and (2.37), the time derivative of γ = 1 − αβ is given by (2.91) γtp = iγ (gp+ ,+ − gp−+ ,+ ) − (gp− ,− − gp−− ,− ) .

(Alternatively, this follows from computing the trace of Utp U −1 = Vp+ − U Vp U −1 .) For instance, if α, β satisfy AL1 (α, β) = 0, then (2.92) γt1 = iγ α(c0,− β + + c0,+ β − ) − β(c0,+ α+ + c0,− α− ) .

Remark 2.12. From (2.10)–(2.13) and the explicit computations of the coefficients fℓ,± , gℓ,± , and hℓ,± , one concludes that the zero-curvature equation (2.82) and hence the Ablowitz–Ladik hierarchy is invariant under the scaling transformation α → αc = {c α(n)}n∈Z ,

β → βc = {β(n)/c}n∈Z ,

c ∈ C \ {0}.

(2.93)

154


Moreover, Rp = G2p − Hp Fp and hence {Em }2p+1 m=0 are invariant under this trans-

formation. Furthermore, choosing c = eicp t , one verifies that it is no restriction to assume cp = 0. This also shows that stationary solutions α, β can only be constructed up to a multiplicative constant. Remark 2.13. odin(i) The special choices β = ±α, c0,± = 1 lead to the discrete nonlinear Schr¨ ger hierarchy. In particular, choosing c(1,1) = −2 yields the discrete nonlinear Schr¨ odinger equation in its usual form (see, e.g., [7, Ch. 3] and the references cited therein), −iαt − (1 ∓ |α|2 )(α− + α+ ) + 2α = 0,

(2.94)

αt − (1 − |α|2 )(α+ − α− ) = 0

(2.95)

as its first nonlinear element. The choice β = α is called the defocusing case, odinger β = −α represents the focusing case of the discrete nonlinear Schr¨ hierarchy. (ii) The alternative choice β = α, c0,± = ∓i, leads to the hierarchy of Schur flows. In particular, choosing c(1,1) = 0 yields as the first nonlinear element of this hierarchy (cf. [10], [24], [25], [36], [41], [50]).

3. The stationary Ablowitz–Ladik formalism This section is devoted to a detailed study of the stationary Ablowitz–Ladik hierarchy. Our principal tools are derived from combining the polynomial recursion formalism introduced in Section 2 and a fundamental meromorphic function φ on a hyperelliptic curve Kp . With the help of φ we study the Baker–Akhiezer vector Ψ, and trace formulas for α and β. Unless explicitly stated otherwise, we suppose in this section that α, β ∈ CZ ,

α(n)β(n) ∈ / {0, 1}, n ∈ Z,

(3.1)

and assume (2.5), (2.6), (2.8), (2.18)–(2.21), (2.32)–(2.39), (2.40), (2.45), (2.69), (2.72), keeping p ∈ N0 fixed. We recall the hyperelliptic curve 2p− Kp : Fp (z, y) = y 2 − 4c−2 Rp (z) = y 2 − 0,+ z

Rp (z) =

c0,+ 2z p−

2 2p+1 m=0

(z − Em ),

2p+1 m=0

(z − Em ) = 0, (3.2)

{Em }2p+1 m=0 ⊂ C \ {0}, p = p− + p+ − 1,

as introduced in (2.75). Throughout this section we assume the affine part of Kp to be nonsingular, that is, we suppose that Em = Em′ for m = m′ , m, m′ = 0, 1, . . . , 2p + 1.

(3.3)


155

Kp is compactified by joining two points P∞± , P∞+ = P∞− , but for notational simplicity the compactification is also denoted by Kp . Points P on Kp \{P∞+ , P∞− } are represented as pairs P = (z, y), where y( · ) is the meromorphic function on Kp satisfying Fp (z, y) = 0. The complex structure on Kp is then defined in the usual manner. Hence, Kp becomes a two-sheeted hyperelliptic Riemann surface of genus p in a standard manner. We also emphasize that by fixing the curve Kp (i.e., by fixing E0 , . . . , E2p+1 ), the summation constants c1,± , . . . , cp± ,± in fp± ,± , gp± ,± , and hp± ,± (and hence in the corresponding stationary s-ALp equations) are uniquely determined as is clear from (2.79), (2.80), which establish the summation constants cℓ,± as symmetric ±1 . functions of E0±1 , . . . , E2p+1 For notational simplicity we will usually tacitly assume that p ∈ N and hence p ∈ N20 \{(0, 0), (0, 1), (1, 0)}. (The trivial case p = 0 is explicitly treated in Example 3.5.) We denote by {µj (n)}j=1,...,p and {νj (n)}j=1,...,p the zeros of ( · )p− Fp ( · , n) and ( · )p− −1 Hp ( · , n), respectively. Thus, we may write Fp (z) = −c0,+ α+ z −p− Hp (z) = c0,+ βz −p− +1

p

j=1 p

j=1

(z − µj ),

(z − νj ),

(3.4)

(3.5)

and we recall that (cf. (2.69)) Rp − G2p = −Fp Hp .

(3.6)

The next step is crucial; it permits us to “lift” the zeros µj and νj from the complex plane C to the curve Kp . From (3.6) one infers that Rp (z) − Gp (z)2 = 0,

z ∈ {µj , νk }j,k=1,...,p .

(3.7)

We now introduce {ˆ µj }j=1,...,p ⊂ Kp and {ˆ νj }j=1,...,p ⊂ Kp by µ ˆj (n) = (µj (n), (2/c0,+ )µj (n)p− Gp (µj (n), n)),

j = 1, . . . , p, n ∈ Z,

(3.8)

νˆj (n) = (νj (n), −(2/c0,+ )νj (n)p− Gp (νj (n), n)),

j = 1, . . . , p, n ∈ Z.

(3.9)

and

We also introduce the points P0,± by P0,± = (0, ±(c0,− /c0,+ )) ∈ Kp ,

2p+1 c20,− = Em . 2 c0,+ m=0

(3.10)

We emphasize that P0,± and P∞± are not necessarily on the same sheet of Kp . Next, we briefly recall our conventions used in connection with divisors on Kp . A map, D : Kp → Z, is called a divisor on Kp if D(P ) = 0 for only finitely

156


many P ∈ Kp . The set of divisors on Kp is denoted by Div(Kp ). We shall employ the following (additive) notation for divisors, DQ0 Q = DQ0 + DQ ,

DQ = DQ1 + · · · + DQm , m

Q0 ∈ Kp , m ∈ N,

Q = {Q1 , . . . , Qm } ∈ Sym Kp , where for any Q ∈ Kp , DQ : Kp → N0 ,

P → DQ (P ) =

/

(3.11)

1 0

for P = Q, for P ∈ Kp \ {Q},

(3.12)

and Symn Kp denotes the nth symmetric product of Kp . In particular, Symm Kp can be identified with the set of nonnegative divisors 0 ≤ D ∈ Div(Kp ) of degree m. Moreover, for a nonzero, meromorphic function f on Kp , the divisor of f is denoted by (f ). Two divisors D, E ∈ Div(Kp ) are called equivalent, denoted by D ∼ E, if and only if D − E = (f ) for some f ∈ M(Kp ) \ {0}. The divisor class [D] of D is then given by [D] = {E ∈ Div(Kp ) | E ∼ D}. We recall that (3.13) deg((f )) = 0, f ∈ M(Kp ) \ {0}, where the degree deg(D) of D is given by deg(D) = P ∈Kp D(P ). Next we introduce the fundamental meromorphic function on Kp by φ(P, n) = =

(c0,+ /2)z −p− y + Gp (z, n)

(3.14)

Fp (z, n) −Hp (z, n)

(c0,+ /2)z −p− y − Gp (z, n)

,

(3.15)

P = (z, y) ∈ Kp , n ∈ Z,

with divisor (φ( · , n)) of φ( · , n) given by (φ( · , n)) = DP0,− νˆ(n) − DP∞− μˆ (n) ,

(3.16)

using (3.4) and (3.5). Here we abbreviated µ1 , . . . , µ ˆp }, νˆ = {ˆ ν1 , . . . , νˆp } ∈ Symp (Kp ). µ ˆ = {ˆ

(3.17)

(The function φ is closely related to one of the variants of Weyl–Titchmarsh functions discussed in [34], [35], [46] in the special defocusing case β = α ¯ .) Given φ( · , n), the meromorphic stationary Baker–Akhiezer vector Ψ( · , n, n0 ) on Kp is then defined by

ψ1 (P, n, n0 ) , Ψ(P, n, n0 ) = ψ2 (P, n, n0 ) ⎧n ′ − ′ ⎪ n ≥ n0 + 1, ⎨ n′ =n0 +1 z + α(n )φ (P, n ) , ψ1 (P, n, n0 ) = 1, (3.18) n = n0 , ⎪ ⎩n0 ′ − ′ −1 , n ≤ n0 − 1, n′ =n+1 z + α(n )φ (P, n )


157

⎧ n ′ − ′ −1 ⎪ n ≥ n0 + 1, ⎨ n′ =n0 +1 zβ(n )φ (P, n ) + 1 , ψ2 (P, n, n0 ) = φ(P, n0 ) 1, n = n0 , ⎪ −1 ⎩n0 ′ − ′ −1 zβ(n )φ (P, n ) + 1 , n ≤ n0 − 1. ′ n =n+1 (3.19)

Basic properties of φ and Ψ are summarized in the following result.

Lemma 3.1. Suppose that α, β satisfy (3.1) and the pth stationary Ablowitz–Ladik system (2.45). Moreover, assume (3.2) and (3.3) and let P = (z, y) ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− }, (n, n0 ) ∈ Z2 . Then φ satisfies the Riccati-type equation αφ(P )φ− (P ) − φ− (P ) + zφ(P ) = zβ,

(3.20)

as well as φ(P )φ(P ∗ ) =

Hp (z) Fp (z)

φ(P ) + φ(P ∗ ) = 2

,

Gp (z) Fp (z)

(3.21) ,

φ(P ) − φ(P ∗ ) = c0,+ z −p−

(3.22) y(P ) . Fp (z)

(3.23)

The vector Ψ satisfies U (z)Ψ− (P ) = Ψ(P ),

(3.24)

Vp (z)Ψ− (P ) = −(i/2)c0,+ z −p− yΨ− (P ),

(3.25)

ψ2 (P, n, n0 ) = φ(P, n)ψ1 (P, n, n0 ), ψ1 (P, n, n0 )ψ1 (P ∗ , n, n0 ) = z n−n0

Fp (z, n)

(3.26) Γ(n, n0 ),

(3.27)

Γ(n, n0 ),

(3.28)

ψ1 (P, n, n0 )ψ2 (P ∗ , n, n0 ) + ψ1 (P ∗ , n, n0 )ψ2 (P, n, n0 )

(3.29)

ψ2 (P, n, n0 )ψ2 (P ∗ , n, n0 ) = z n−n0

= 2z n−n0

Gp (z, n) Fp (z, n0 )

Fp (z, n0 ) Hp (z, n) Fp (z, n0 )

Γ(n, n0 ),

ψ1 (P, n, n0 )ψ2 (P ∗ , n, n0 ) − ψ1 (P ∗ , n, n0 )ψ2 (P, n, n0 ) y Γ(n, n0 ), = −c0,+ z n−n0 −p− Fp (z, n0 ) where we used the abbreviation ⎧n ′ ⎪ ⎨ n′ =n0 +1 γ(n ), Γ(n, n0 ) = 1, ⎪ ⎩n0 ′ −1 , n′ =n+1 γ(n )

n ≥ n0 + 1, n = n0 , n ≤ n0 − 1.

(3.30)

(3.31)

158


Proof. To prove (3.20) one uses the definition (3.14) of φ and equations (2.10), (2.12), and (2.69) to obtain αφ(P )φ− (P ) − φ(P )− + zφ(P ) − zβ 1 −p− αGp G− = yα(Gp + G− p + (c0,+ /2)z p ) + αRp Fp Fp−

. −p− − (G− y)Fp + z(Gp + (c0,+ /2)z −p− y)Fp− − zβFp Fp− p + (c0,+ /2)z . 1 − − − − = − αGp (Gp + Gp ) + Fp (−αHp − Gp − zβFp ) + zFp Gp = 0. (3.32) Fp Fp

Equations (3.21)–(3.23) are clear from the definitions of φ and y. By definition of ψ, (3.26) holds for n = n0 . By induction, zβ(n)φ− (P, n)−1 + 1 ψ2− (P, n, n0 ) ψ2 (P, n, n0 ) zβ(n) + φ− (P, n) = = , ψ1 (P, n, n0 ) z + α(n)φ− (P, n) ψ1− (P, n, n0 ) z + α(n)φ− (P, n)

(3.33)

and hence ψ2 /ψ1 satisfies the Riccati-type equation (3.20) α(n)φ− (P, n)

ψ2 (P, n, n0 ) ψ2 (P, n, n0 ) − φ− (P, n) + z − zβ(n) = 0. ψ1 (P, n, n0 ) ψ1 (P, n, n0 )

(3.34)

This proves (3.26). The definition of ψ implies ψ1 (P, n, n0 ) = (z + α(n)φ− (P, n))ψ1− (P, n, n0 ) = zψ1− (P, n, n0 ) + α(n)ψ2− (P, n, n0 ),

(3.35)

ψ2 (P, n, n0 ) = (zβ(n)φ− (P, n)−1 + 1)ψ2− (P, n, n0 ) = zβ(n)ψ1− (P, n, n0 ) + ψ2− (P, n, n0 ),

(3.36)

which proves (3.24). Property (3.25) follows from (3.26) and the definition of φ. To prove (3.27) one can use (2.10) and (2.12) ψ1 (P )ψ1 (P ∗ ) = (z + αφ− (P ))(z + αφ− (P ∗ ))ψ1− (P )ψ1− (P ∗ ) 1 − − 2 − ∗ = − (z 2 Fp− + 2zαG− p + α Hp )ψ1 (P )ψ1 (P ) Fp 1 − − ∗ = − (z 2 Fp− − zαβFp + zα(Gp + G− p ))ψ1 (P )ψ1 (P ) Fp = zγ

Fp Fp−

ψ1− (P )ψ1− (P ∗ ).

(3.37)

Equation (3.28) then follows from (3.22) and (3.24). Finally, equation (3.29) (resp. (3.30)) is proved by combining (3.22) and (3.26) (resp. (3.23) and (3.26)). Combining the Laurent polynomial recursion approach of Section 2 with (3.4) and (3.5) readily yields trace formulas for fℓ,± and hℓ,± in terms of symmetric


159

functions of the zeros µj and νk of ( · )p− Fp and ( · )p− −1 Hp , respectively. For simplicity we just record the simplest cases. Lemma 3.2. Suppose that α, β satisfy (3.1) and the pth stationary Ablowitz–Ladik system (2.45). Then, 2p+1

−1/2 p α p+1 , (3.38) = (−1) µ E j m α+ m=0 j=1

−1/2 2p+1 p β+ = (−1)p+1 , Em νj β m=0 j=1

p

j=1 p j=1

µj = α+ β − γ + νj = α+ β − γ

α++ c1,+ − , α+ c0,+

(3.39)

(3.40)

c1,+ β− . − β c0,+

(3.41)

Proof. We compare coefficients in (2.18) and (3.4) z p− Fp (z) = f0,− + · · · + z p− +p+ −2 f1,+ + z p− +p+ −1 f0,+

p p = c0,+ α+ (−1)p+1 µj + · · · + z p− +p+ −2 µj − z p− +p+ −1 j=1

j=1

(3.42) + 2 + ++ + − α c1,+ which yields and use f0,− = c0,− α and f1,+ = c0,+ (α ) β − γ α (3.38) and (3.40). Similarly, one employs h0,− = −c0,− β + and h1,+ = c0,+ γβ − − α+ β 2 + βc1,+ for the remaining formulas (3.39) and (3.41).

Next we turn to asymptotic properties of φ and Ψ in a neighborhood of P∞± and P0,± .

Lemma 3.3. Suppose that α, β satisfy (3.1) and the pth stationary Ablowitz–Ladik system (2.45). Moreover, let P = (z, y) ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− }, (n, n0 ) ∈ Z2 . Then φ has the asymptotic behavior / P → P∞+ , β + β − γζ + O(ζ 2 ), φ(P ) = ζ = 1/z, ζ→0 −(α+ )−1 ζ −1 + (α+ )−2 α++ γ + + O(ζ), P → P∞− , (3.43) / −1 −2 − 2 P → P0,+ , α − α α γζ + O(ζ ), ζ = z. (3.44) φ(P ) = ζ→0 −β + ζ − β ++ γ + ζ 2 + O(ζ 3 ), P → P0,− ,

The components of the Baker–Akhiezer vector Ψ have the asymptotic behavior / ζ n0 −n (1 + O(ζ)), P → P∞+ , ψ1 (P, n, n0 ) = ζ = 1/z, (3.45) α+ (n) ζ→0 Γ(n, n ) + O(ζ), P → P∞− , 0 α+ (n0 )

160


ψ1 (P, n, n0 ) =

ζ→0

/

ψ2 (P, n, n0 ) =

/

ψ2 (P, n, n0 ) =

/

ζ→0

ζ→0

α(n) α(n0 ) + n−n0

ζ

O(ζ),

Γ(n, n0 )(1 + O(ζ)),

P → P0,+ ,

P → P0,− ,

β(n)ζ n0 −n (1 + O(ζ)), 1 − α+ (n Γ(n, n0 )ζ −1 (1 + O(ζ)), 0) 1 α(n0 ) +

ζ = z,

P → P∞+ , P → P∞− ,

+ O(ζ),

−β (n)Γ(n, n0 )ζ n+1−n0 (1 + O(ζ)),

(3.46) ζ = 1/z,

P → P0,+ ,

P → P0,− ,

(3.47) ζ = z. (3.48)

The divisors (ψj ) of ψj , j = 1, 2, are given by (ψ1 ( · , n, n0 )) = Dμˆ (n) − Dμˆ (n0 ) + (n − n0 )(DP0,− − DP∞+ ),

(3.49)

(ψ2 ( · , n, n0 )) = Dνˆ(n) − Dμˆ (n0 ) + (n − n0 )(DP0,− − DP∞+ ) + DP0,− − DP∞− . (3.50)

Proof. The existence of the asymptotic expansion of φ in terms of the local coordinate ζ = 1/z near P∞± , respectively, ζ = z near P0,± is clear from the explicit form of φ in (3.14) and (3.15). Insertion of the Laurent polynomials Fp into (3.14) and Hp into (3.15) then yields the explicit expansion coefficients in (3.43) and (3.44). Alternatively, and more efficiently, one can insert each of the following asymptotic expansions φ(P ) = φ−1 z + φ0 + φ1 z −1 + O(z −2 ), z→∞

φ(P ∗ ) = φ0 + φ1 z −1 + O(z −2 ), z→∞

φ(P ) = φ0 + φ1 z + O(z 2 ),

(3.51)

z→0

φ(P ∗ ) = φ1 z + φ2 z 2 + O(z 3 ) z→0

into the Riccati-type equation (3.20) and, upon comparing coefficients of powers of z, which determines the expansion coefficients φk in (3.51), one concludes (3.43) and (3.44). Next we compute the divisor of ψ1 . By (3.18) it suffices to compute the divisor of z + αφ− (P ). First of all we note that ⎧ z + O(1), P → P∞+ , ⎪ ⎪ ⎨ α+ −1 γ + O(z ), P → P∞− , − α (3.52) z + αφ (P ) = α ⎪ P → P0,+ , − + O(z), ⎪ α ⎩ 2 γz + O(z ), P → P0,− ,

which establishes (3.45) and (3.46). Moreover, the poles of the function z +αφ− (P ) in Kp \ {P0,± , P∞± } coincide with the ones of φ− (P ), and so it remains to compute the missing p zeros in Kp \ {P0,± , P∞± }. Using (2.12), (2.21), (2.69), and y(ˆ µj ) =


161

p

(2/c0,+ )µj − Gp (µj ) (cf. (3.8)) one computes z + αφ− (P ) = z + α = =

(c0,+ /2)z −p− y + G− p

Fp + α((c0,+ /2)z Fp− Fp Fp−

+α

Fp− −p−

y − Gp )

(c0,+ /2)2 z −2p− y 2 − G2p

Fp− ((c0,+ /2)z −p− y + Gp )

αHp Fp (z) Fp = = − 1+ O(1). (c0,+ /2)z −p− y + Gp P →ˆμj Fp− (z) Fp

(3.53)

Hence the sought after zeros are at µ ˆ j , j = 1, . . . , p (with the possibility that a ˆ− zero at µ ˆ j is cancelled by a pole at µ j ). Finally, the behavior of ψ2 follows immediately using ψ2 = φψ1 . In addition to (3.43), (3.44) one can use the Riccati-type equation (3.20) to derive a convergent expansion of φ around P∞± and P0,± and recursively determine the coefficients as in Lemma 3.3. Since this is not used later in this section, we omit further details at this point. Since nonspecial divisors play a fundamental role in the derivation of theta function representations of algebro-geometric solutions of the AL hierarchy in [31], we now take a closer look at them. Lemma 3.4. Suppose that α, β satisfy (3.1) and the pth stationary Ablowitz–Ladik ˆ = system (2.45). Moreover, assume (3.2) and (3.3) and let n ∈ Z. Let Dμˆ , µ ˆp }, and Dνˆ , νˆ = {ˆ ν 1 , . . . , νˆp }, be the pole and zero divisors of degree p, {ˆ µ1 , . . . , µ respectively, associated with α, β, and φ defined according to (3.8) and (3.9), that is, µ ˆj (n) = (µj (n), (2/c0,+ )µj (n)p− Gp (µj (n), n)), νˆj (n) = (νj (n), −(2/c0,+ )νj (n)p− Gp (νj (n), n)),

j = 1, . . . , p, j = 1, . . . , p.

(3.54)

Then Dμˆ (n) and Dνˆ(n) are nonspecial for all n ∈ Z.

Proof. We provide a detailed proof in the case of Dμˆ (n) . By [30, Thm. A.31] (see also [29, Thm. A.30]), Dμˆ (n) is special if and only if {ˆ µ1 (n), . . . , µ ˆp (n)} contains at least one pair of the type {ˆ µ(n), µ ˆ(n)∗ }. Hence Dμˆ (n) is certainly nonspecial as long as the projections µj (n) of µ ˆj (n) are mutually distinct, µj (n) = µk (n) for j = k. On the other hand, if two or more projections coincide for some n0 ∈ Z, for instance, (3.55) µj1 (n0 ) = · · · = µjN (n0 ) = µ0 , N ∈ {2, . . . , p},

/ {E0 , . . . , E2p+1 }. This fact immediately follows then Gp (µ0 , n0 ) = 0 as long as µ0 ∈ from (2.69) since Fp (µ0 , n0 ) = 0 but Rp (µ0 ) = 0 by hypothesis. In particular,

162


µ ˆj1 (n0 ), . . . , µ ˆjN (n0 ) all meet on the same sheet since p

µ ˆjr (n0 ) = (µ0 , (2/c0,+ )µ0− Gp (µ0 , n0 )),

r = 1, . . . , N,

(3.56)

and hence no special divisor can arise in this manner. Remaining to be studied is the case where two or more projections collide at a branch point, say at (Em0 , 0) for some n0 ∈ Z. In this case one concludes Fp (z, n0 ) = O (z − Em0 )2 and z→Em0

Gp (Em0 , n0 ) = 0

(3.57)

using again (2.69) and Fp (Em0 , n0 ) = Rp (Em0 ) = 0. Since Gp ( · , n0 ) is a Laurent polynomial, (3.57) implies Gp (z, n0 ) = O((z − Em0 )). Thus, using (2.69) once z→Em0

more, one obtains the contradiction, = Rp (z) O (z − Em0 )2 z→Em0

=

z→Em0

c0,+ p 2Em−0

2

(3.58) 0

(z − Em0 )

2p+1

m=0 m =m0

1

Em0 − Em + O(z − Em0 ) .

Consequently, at most one µ ˆj (n) can hit a branch point at a time and again no special divisor arises. Finally, by our hypotheses on α, β, µ ˆ j (n) stay finite for fixed n ∈ Z and hence never reach the points P∞± . (Alternatively, by (3.43), µ ˆj never ˆ j tend to infinity, they all necessarily reaches the point P∞+ . Hence, if some µ converge to P∞− .) Again no special divisor can arise in this manner. The proof for Dνˆ(n) is analogous (replacing Fp by Hp and noticing that by (3.43), φ has no zeros near P∞± ), thereby completing the proof. The results of Sections 2 and 3 have been used extensively in [31] to derive the class of stationary algebro-geometric solutions of the Ablowitz–Ladik hierarchy and the associated theta function representations of α, β, φ, and Ψ. These theta function representations also show that γ(n) ∈ / {0, 1} for all n ∈ Z, and hence condition (3.1) is satisfied for the stationary algebro-geometric AL solutions discussed in this section, provided the associated divisors Dμˆ (n) and Dνˆ(n) stay away from P∞± , P0,± for all n ∈ Z. We conclude this section with the trivial case p = 0 excluded thus far. Example 3.5. Assume p = 0 and c0,+ = c0,− = c0 = 0 (we recall that gp+ ,+ = gp− ,− ). Then, * (0,0) = 0, F(0,0) = F*(0,0) = H(0,0) = H

G(0,0) = K(0,0) =

* (0,0) = K * (0,0) = 1 , R(0,0) = 1 c20 , G 2 4 α = β = 0,

ic0 1 0 z 0 . , V(0,0) = U= 0 1 2 0 −1

1 c0 , 2

(3.59)

The Ablowitz–Ladik Hierarchy Revisited Introducing Ψ+ (z, n, n0 ) =

n−n 0 z , 0

Ψ− (z, n, n0 ) =

one verifies the equations

U Ψ− ± = Ψ± ,

0 , 1

V(0,0) Ψ− ± = ±

n, n0 ∈ Z,

ic0 − Ψ . 2 ±

163

(3.60)

(3.61)

4. The time-dependent Ablowitz–Ladik formalism In this section we extend the algebro-geometric analysis of Section 3 to the timedependent Ablowitz–Ladik hierarchy. For most of this section we assume the following hypothesis. Hypothesis 4.1. (i) Suppose that α, β satisfy α( · , t), β( · , t) ∈ CZ , t ∈ R,

α(n, · ), β(n, · ) ∈ C 1 (R), n ∈ Z,

α(n, t)β(n, t) ∈ / {0, 1}, (n, t) ∈ Z × R.

(4.1)

(ii) Assume that the hyperelliptic curve Kp satisfies (3.2) and (3.3). The fundamental problem in the analysis of algebro-geometric solutions of the Ablowitz–Ladik hierarchy consists of solving the time-dependent rth Ablowitz– Ladik flow with initial data a stationary solution of the pth system in the hierarchy. More precisely, given p ∈ N20 \ {(0, 0)} we consider a solution α(0) , β (0) of the pth stationary Ablowitz–Ladik system s-ALp (α(0) , β (0) ) = 0, associated with the hyperelliptic curve Kp and a corresponding set of summation constants {cℓ,± }ℓ=1,...,p± ⊂ C. Next, let r = (r− , r+ ) ∈ N20 ; we intend to construct a solution α, β of the rth Ablowitz–Ladik flow ALr (α, β) = 0 with α(t0,r ) = α(0) , β(t0,r ) = β (0) for some t0,r ∈ R. To emphasize that the summation constants in the definitions of the stationary and the time-dependent Ablowitz–Ladik equations are independent of each other, we indicate this by adding a tilde on all the )r , H )r , time-dependent quantities. Hence we shall employ the notation V)r , F)r , G ˜ s,± , c˜s,± , in order to distinguish them from Vp , Fp , Gp , Hp , Kp , ) r , f˜s,± , g˜s,± , h K fℓ,± , gℓ,± , hℓ,± , cℓ,± , in the following. In addition, we will follow a more elaborate notation inspired by Hirota’s τ -function approach and indicate the individual rth Ablowitz–Ladik flow by a separate time variable tr ∈ R. Summing up, we are interested in solutions α, β of the time-dependent algebro-geometric initial value problem 1 0 gr+ ,+ + g˜r−− ,− ) + f˜r+ −1,+ − f˜r−− −1,− r (α, β) = −iαtr − α(˜ AL = 0, ˜ r −1,− + h ˜− gr−+ ,+ + g˜r− ,− ) − h −iβtr + β(˜ − r+ −1,+ (4.2) (α, β) = α(0) , β (0) , t=t0,r

164


s-ALp α(0) , β (0) =

1 0 −α(0) (gp+ ,+ + gp−− ,− ) + fp+ −1,+ − fp−− −1,− =0 − − β (0) (gp+ ,+ + gp− ,− ) − hp− −1,− + hp+ −1,+

(4.3)

for some t0,r ∈ R, where α = α(n, tr ), β = β(n, tr ) satisfy (4.1) and a fixed curve Kp is associated with the stationary solutions α(0) , β (0) in (4.3). Here, p = (p− , p+ ) ∈ N20 \ {(0, 0)},

r = (r− , r+ ) ∈ N20 ,

p = p− + p+ − 1.

(4.4)

In terms of the zero-curvature formulation this amounts to solving Utr (z, tr ) + U (z, tr )V)r (z, tr ) − V)r+ (z, tr )U (z, tr ) = 0,

(4.5)

U (z, t0,r )Vp (z, t0,r ) − Vp+ (z, t0,r )U (z, t0,r ) = 0.

(4.6)

One can show (cf. [32]) that the stationary Ablowitz–Ladik system (4.6) is actually satisfied for all times tr ∈ R: Thus, we actually impose Utr (z, tr ) + U (z, tr )V)r (z, tr ) − V)r+ (z, tr )U (z, tr ) = 0,

(4.7)

U (z, tr )Vp (z, tr ) − Vp+ (z, tr )U (z, tr ) = 0,

(4.8)

instead of (4.5) and (4.6). For further reference, we recall the relevant quantities here (cf. (2.5), (2.6), (2.18)–(2.22)):

z α , U (z) = zβ 1 1 1 0 − 0 − (4.9) ) (z) −F) − (z) Gp (z) −Fp− (z) G r r ) Vp (z) = i , , Vr (z) = i − − ) − (z) ) − (z) −K Hp (z) −Gp (z) H r r and

Fp (z) =

p− ℓ=1 p−

Gp (z) =

Hp (z) =

fp− −ℓ,− z −ℓ + gp− −ℓ,− z −ℓ +

ℓ=1 p− −1

ℓ=0

F)r (z) =

) r (z) = G

) r (z) = H

r− s=1 r−

g˜r− −s,− z −s +

s=0

ℓ=0 p+

ℓ=0

hp− −1−ℓ,− z −ℓ +

f˜r− −s,− z −s +

s=1 r− −1

p+ −1

p+ ℓ=1

s=0 r+

s=0

˜ r− −1−s,− z −s + h

p

j=1

(z − µj ),

gp+ −ℓ,+ z ℓ ,

r+ −1

fp+ −1−ℓ,+ z ℓ = −c0,+ α+ z −p−

hp+ −ℓ,+ z ℓ = c0,+ βz −p− +1

f˜r+ −1−s,+ z s ,

g˜r+ −s,+ z s , r+ s=1

˜ r+ −s,+ z s , h

p

j=1

(z − νj ), (4.10)

The Ablowitz–Ladik Hierarchy Revisited ) r (z) = K

r− s=0

g˜r− −s,− z −s +

r+ s=1

165

)r (z) + g˜r− ,− − g˜r+ ,+ g˜r+ −s,+ z s = G

(4.11)

˜ s,± for fixed p ∈ N20 \ {(0, 0)}, r ∈ N20 . Here fℓ,± , f˜s,± , gℓ,± , g˜s,± , hℓ,± , and h are defined as in (2.32)–(2.39) with appropriate sets of summation constants cℓ,± , ℓ ∈ N0 , and c˜k,± , k ∈ N0 . Explicitly, (4.7) and (4.8) are equivalent to (cf. (2.10)– (2.13), (2.83)–(2.86)), )r + K ) r− ) − F)r , (4.12) αtr = i z F)r− + α(G )− + K )r) − H ) r + z −1 H )− , (4.13) βtr = −i β(G r r ) r ) + zβ F)r + αH ) −, )− − G 0 = z(G r r

(4.14)

)r + K ) r− − K )r, 0 = zβ F)r− + αH

(4.15)

− 0 = z(G− p − Gp ) + zβFp + αHp ,

0=

zβFp−

+ αHp − Gp +

(4.16)

G− p,

(4.17)

0 = −Fp + zFp− + α(Gp + G− p ),

(4.18)

− 0 = zβ(Gp + G− p ) − zHp + Hp ,

(4.19)

respectively. In particular, (2.69) holds in the present tr -dependent setting, that is, G2p − Fp Hp = Rp .

(4.20)

As in the stationary context (3.8), (3.9) we introduce µ ˆj (n, tr ) = (µj (n, tr ), (2/c0,+ )µj (n, tr )p− Gp (µj (n, tr ), n, tr )) ∈ Kp ,

(4.21)

νˆj (n, tr ) = (νj (n, tr ), −(2/c0,+ )νj (n, tr )p− Gp (νj (n, tr ), n, tr )) ∈ Kp ,

(4.22)

j = 1, . . . , p, (n, tr ) ∈ Z × R,

and j = 1, . . . , p, (n, tr ) ∈ Z × R,

and note that the regularity assumptions (4.1) on α, β imply continuity of µj and νk with respect to tr ∈ R (away from collisions of these zeros, µj and νk are of course C ∞ ). In analogy to (3.14), (3.15), one defines the following meromorphic function φ( · , n, tr ) on Kp , φ(P, n, tr ) = =

(c0,+ /2)z −p− y + Gp (z, n, tr )

(4.23)

Fp (z, n, tr ) −Hp (z, n, tr )

(c0,+ /2)z −p− y − Gp (z, n, tr )

,

P = (z, y) ∈ Kp , (n, tr ) ∈ Z × R,

(4.24)

166


with divisor (φ( · , n, tr )) of φ( · , n, tr ) given by (φ( · , n, tr )) = DP0,− νˆ(n,tr ) − DP∞− μˆ (n,tr ) .

(4.25)

The time-dependent Baker–Akhiezer vector is then defined in terms of φ by

ψ1 (P, n, n0 , tr , t0,r ) , (4.26) Ψ(P, n, n0 , tr , t0,r ) = ψ2 (P, n, n0 , tr , t0,r ) tr

)r (z, n0 , s) − F)r (z, n0 , s)φ(P, n0 , s) ψ1 (P, n, n0 , tr , t0,r ) = exp i ds G t0,r

⎧ n ′ − ′ ⎪ n ≥ n0 + 1, ⎨ n′ =n0 +1 z + α(n , tr )φ (P, n , tr ) , × 1, (4.27) n = n0 , ⎪ −1 ⎩n0 ′ − ′ z + α(n , t )φ (P, n , t ) , n ≤ n − 1, ′ r r 0 n =n+1 tr

)r (z, n0 , s) − F)r (z, n0 , s)φ(P, n0 , s) ds G ψ2 (P, n, n0 , tr , t0,r ) = exp i t0,r

⎧ n ′ − ′ −1 ⎪ n ≥ n0 + 1, ⎨ n′ =n0 +1 zβ(n , tr )φ (P, n , tr ) + 1 , × φ(P, n0 , tr ) 1, n = n0 , ⎪ −1 ⎩n0 − ′ −1 ′ +1 , n ≤ n0 − 1, n′ =n+1 zβ(n , tr )φ (P, n , tr ) (4.28) P = (z, y) ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− }, (n, tr ) ∈ Z × R.

One observes that ψ1 (P, n, n0 , tr , t˜r ) = ψ1 (P, n0 , n0 , tr , t˜r )ψ1 (P, n, n0 , tr , tr ), P = (z, y) ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− }, (n, n0 , tr , t˜r ) ∈ Z2 × R2 .

(4.29)

The following lemma records basic properties of φ and Ψ in analogy to the stationary case discussed in Lemma 3.1. Lemma 4.2. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. In addition, let P = (z, y) ∈ Kp \ {P∞+ , P∞− }, (n, n0 , tr , t0,r ) ∈ Z2 × R2 . Then φ satisfies αφ(P )φ− (P ) − φ− (P ) + zφ(P ) = zβ, )r (z) + K ) r (z) φ(P ) + iH ) r (z), φtr (P ) = iF)r φ2 (P ) − i G φ(P )φ(P ∗ ) =

Hp (z) Fp (z)

φ(P ) + φ(P ∗ ) = 2

,

Gp (z) Fp (z)

,

φ(P ) − φ(P ∗ ) = c0,+ z −p−

(4.30) (4.31) (4.32) (4.33)

y(P ) . Fp (z)

(4.34)


167

Moreover, assuming P = (z, y) ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− }, then Ψ satisfies ψ2 (P, n, n0 , tr , t0,r ) = φ(P, n, tr )ψ1 (P, n, n0 , tr , t0,r ),

(4.35)

U (z)Ψ− (P ) = Ψ(P ),

(4.36)

Vp (z)Ψ− (P ) = −(i/2)c0,+z −p− yΨ− (P ),

(4.37)

Ψtr (P ) = V)r+ (z)Ψ(P ),

(4.38)

ψ1 (P, n, n0 , tr , t0,r )ψ1 (P ∗ , n, n0 , tr , t0,r ) = z n−n0

ψ2 (P, n, n0 , tr , t0,r )ψ2 (P ∗ , n, n0 , tr , t0,r ) = z n−n0 ∗

Fp (z, n, tr ) Fp (z, n0 , t0,r ) Hp (z, n, tr ) Fp (z, n0 , t0,r )

Γ(n, n0 , tr ), (4.39) Γ(n, n0 , tr ), (4.40)

∗

ψ1 (P, n, n0 , tr , t0,r )ψ2 (P , n, n0 , tr , t0,r ) + ψ1 (P , n, n0 , tr , t0,r )ψ2 (P, n, n0 , tr , t0,r ) = 2z n−n0

Gp (z, n, tr ) Fp (z, n0 , t0,r )

Γ(n, n0 , tr ),

(4.41)

ψ1 (P, n, n0 , tr , t0,r )ψ2 (P ∗ , n, n0 , tr , t0,r ) − ψ1 (P ∗ , n, n0 , tr , t0,r )ψ2 (P, n, n0 , tr , t0,r ) y Γ(n, n0 , tr ), = −c0,+ z n−n0 −p− (4.42) Fp (z, n0 , t0,r ) where

⎧n ′ ⎪ ⎨ n′ =n0 +1 γ(n , tr ), Γ(n, n0 , tr ) = 1, ⎪ ⎩n0 ′ −1 , n′ =n+1 γ(n , tr )

n ≥ n0 + 1, n = n0 , n ≤ n0 − 1.

(4.43)

In addition, as long as the zeros µj (n0 , s) of ( · )p− Fp ( · , n0 , s) are all simple and distinct from zero for s ∈ Iμ , Iμ ⊆ R an open interval, Ψ( · , n, n0 , tr , t0,r ) is meromorphic on Kp \ {P∞+ , P∞− , P0,+ , P0,− } for (n, tr , t0,r ) ∈ Z × Iμ2 . Proof. Equations (4.30), (4.32)–(4.37), and (4.39)–(4.42) are proved as in the stationary case, see Lemma 3.1. Thus, we turn to the proof of (4.31) and (4.38): Differentiating the Riccati-type equation (4.30) yields 0 = αφφ− − φ− + zφ − zβ tr = αtr φφ− + (αφ− + z)φtr + (αφ − 1)φ− tr − zβtr − − − ) − ) + z F) − − F)r )r + K = (αφ + z) + (αφ − 1)S φtr + iφφ α(G r r ) r ) + i(z H )r − H ) − ), )− + K + izβ(G r r

(4.44)

using (4.12) and (4.13). Next, one employs (3.20) to rewrite (αφ− + z) + (αφ − 1)S − =

z 1 (zβ + φ− ) + − (β − φ)S − . φ φ

(4.45)

168


This allows one to calculate the right-hand side of (4.31) using (4.14) and (4.15) )r + K ) r )φ ) r + F)r φ2 − (G (αφ− + z) + (αφ − 1)S − H ) r + (αφ − 1)H ) r− + φ(zβ + φ− )F)r + zφ− (β − φ)F)r− = (αφ− + z)H )r + K ) r ) − z(β − φ)(G )− + K ) −) − (zβ + φ− )(G r r

)r − H ) − + φ− (αH ) r + zβ F) − ) + φ(αH ) − + zβ F)r ) = φφ− (F)r − z F)r− ) + z H r r r − ) )r + G )− )− )− )− ) )r + K − zβ(G r + Kr ) − zφ(Gr + Kr ) − φ (Gr + Kr )

− )r − H ) r− − zβ(G )− ) ) )− = φφ− (F)r − z F)r− ) + z H r + Kr ) + (zφ − φ − zβ)(Gr + Kr ) − ) )r − H ) r− − zβ(G )− ) )− = φφ− (F)r − z F)r− ) + z H r + Kr ) − αφφ (Gr + Kr ).

(4.46)

Hence, -1 . z ) r − iF)r φ2 + i(G )r + K ) r )φ = 0. (4.47) (zβ + φ− ) + − (β − φ)S − φtr − iH φ φ Solving the first-order difference equation (4.47) then yields φtr (P, n, tr ) − iF)r (z, n, tr )φ(P, n, tr )2

) r (z, n, tr ))φ(P, n, tr ) − iH ) r (z, n, tr ) )r (z, n, tr ) + K + i(G ⎧ n ′ ′ ⎪ n ≥ 1, ⎨ n′ =1 B(P, n , tr )/A(P, n , tr ), = C(P, tr ) 1, n = 0, ⎪ ⎩ 0 ′ ′ n′ =n+1 A(P, n , tr )/B(P, n , tr ), n ≤ −1

(4.48)

for some n-independent function C( · , tr ) meromorphic on Kp , where A = φ−1 (zβ + φ− ),

B = −z(φ− )−1 (β − φ).

(4.49)

The asymptotic behavior of φ(P, n, tr ) in (3.43) then yields (for tr ∈ R fixed) B(P ) A(P )

=

P →P∞+

−(1 − αβ)(β − )−1 z −1 + O(z −2 ).

(4.50)

Since the left-hand side of (4.48) is of order O(z r+ ) as P → P∞+ , and C is meromorphic, insertion of (4.50) into (4.48), taking n ≥ 1 sufficiently large, then yields a contradiction unless C = 0. This proves (4.31). Proving (4.38) is equivalent to showing ) r − φF)r )ψ1 , ψ1,tr = i(G

ψ1 φtr + φψ1,tr

) r − φK ) r )ψ1 , = i(H

using (4.35). Equation (4.52) follows directly from (4.51) and from (4.31), ) r + iF)r φ2 − i(G )r + K ) r )φ + i(G )r − φF)r )φ ψ1 φtr + φψ1,tr = ψ1 iH ) r )ψ1 . ) r − φK = i(H

(4.51) (4.52)

(4.53)


169

To prove (4.51) we start from (z + αφ− )tr = αtr φ− + αφ− tr − − − ) + F)− (φ− )2 − (G ) ) ) )− + K ) − )φ− = φ i z Fr + α(Gr + Kr− ) − F)r + αi H r r r r )− ) + i(z + αφ− )φ− F) − − iφ− F)r + iαH )− )r − G = iαφ− (G r r r − )− ) r − φF)r − (G )− = i(z + αφ− ) G r − φ Fr ) ,

(4.54)

where we used (4.14) and (3.20) to rewrite

Abbreviating

−) )r − G )− ) ) r− − iφ− F)r = iz(G iαH r ) − αφφ Fr − zφFr .

σ(P, n0 , tr ) = i

tr

0

) r (z, n0 , s) − F)r (z, n0 , s)φ(P, n0 , s) , ds G

one computes for n ≥ n0 + 1,

n ψ1,tr = exp(σ) (z + αφ− )(n′ ) n′ =n0 +1

= σtr ψ1 + exp(σ)

n

n

n′ =n0 +1

) r − F)r φ)ψ1 . = i(G

(4.56)

tr

(z + αφ− )tr (n′ )

n′ =n0 +1

= ψ1 σtr + i

(4.55)

n

(z + αφ− )(n′′ )

n′′ =1 ′′ ′

n =n

)r − F)r φ)(n′ − 1) )r − F)r φ)(n′ ) − (G (G

(4.57)

The case n ≤ n0 is handled analogously establishing (4.51). That Ψ( · , n, n0 , tr , t0,r ) is meromorphic on Kp \ {P∞± , P0,± } if Fp ( · , n0 , tr ) has only simple zeros distinct from zero is a consequence of (4.27), (4.28), and of −iF)r (z, n0 , s)φ(P, n0 , s) = ∂s ln Fp (z, n0 , s) + O(1), (4.58) P →ˆ μj (n0 ,s)

using (4.21), (4.25), and (4.59). (Equation (4.59) in Lemma 4.3 follows from (4.31), (4.33), and (4.34) which have already been proven.) Next we consider the tr -dependence of Fp , Gp , and Hp . Lemma 4.3. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. In addition, let (z, n, tr ) ∈ C × Z × R. Then, )r + K ) r Fp , Fp,tr = −2iGp F)r + i G (4.59) ) r − iHp F)r , Gp,tr = iFp H )r + K ) r Hp . )r − i G Hp,tr = 2iGp H

(4.60)

(4.61)

170


In particular, (4.59)–(4.61) are equivalent to Vp,tr = V)r , Vp .

(4.62)

Proof. To prove (4.59) one first differentiates equation (4.34) φtr (P ) − φtr (P ∗ ) = −c0,+ z −p− yFp−2 Fp,tr .

(4.63)

The time derivative of φ given in (4.31) and (4.33) yield ) r + F)r φ(P )2 − G )r + K ) r φ(P ) φtr (P ) − φtr (P ∗ ) = i H )r + K ) r φ(P ∗ ) ) r + F)r φ(P ∗ )2 − G −i H

and hence

= iF)r (φ(P ) + φ(P ∗ ))(φ(P ) − φ(P ∗ )) ) r (φ(P ) − φ(P ∗ )) )r + K −i G ) r yFp−1 , (4.64) )r + K = 2ic0,+ z −p− F)r yGp Fp−2 − ic0,+ z −p− G )r + K ) r Fp . Fp,tr = −2iGp F)r + i G

(4.65)

Similarly, starting from (4.33)

φtr (P ) + φtr (P ∗ ) = 2Fp−2 (Fp Gp,tr − Fp,tr Gp )

(4.66)

0 = Rp,tr = 2Gp Gp,tr − Fp,tr Hp − Fp Hp,tr

(4.67)

yields (4.60) and

proves (4.61).

Next we turn to the Dubrovin equations for the time variation of the zeros r flow. µj of ( · )p− Fp and νj of ( · )p− −1 Hp governed by the AL

Lemma 4.4. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold on Z × Iμ with Iμ ⊆ R an open interval. In addition, assume that the zeros µj , j = 1, . . . , p, of ( · )p− Fp ( · ) remain distinct and nonzero on Z×Iμ . Then {ˆ µj }j=1,...,p , defined in (4.21), satisfies the following first-order system of differential equations on Z × Iμ ,

with

µj,tr = −iF)r (µj )y(ˆ µj )(α+ )−1

p

k=1 k =j

(µj − µk )−1 ,

j = 1, . . . , p,

(4.68)

(4.69) µ ˆj (n, ·) ∈ C ∞ (Iμ , Kp ), j = 1, . . . , p, n ∈ Z. For the zeros νj , j = 1, . . . , p, of ( · )p− −1 Hp ( · ), identical statements hold with µj and Iμ replaced by νj and Iν , etc. (with Iν ⊆ R an open interval ). In particular, {ˆ νj }j=1,...,p , defined in (4.22), satisfies the first-order system on Z × Iν , ) r (νj )y(ˆ νj,tr = iH νj )(βνj )−1

p

k=1 k =j

(νj − νk )−1 ,

j = 1, . . . , p,

(4.70)


171

with νˆj (n, ·) ∈ C ∞ (Iν , Kp ),

j = 1, . . . , p, n ∈ Z.

(4.71)

Proof. It suffices to consider (4.68) for µj,tr . Using the product representation for Fp in (4.10) and employing (4.21) and (4.59), one computes 1 0 p −p Fp,tr (µj ) = c0,+ α+ µj − (µj − µk ) µj,tr = −2iGp (µj )F)r (µj ) k=1 (4.72) −p−

= −ic0,+ µj

k =j

y(ˆ µj )F)r (µj ),

j = 1, . . . , p,

proving (4.68). The case of (4.70) for νj,tr is of course analogous using the product representation for Hp in (4.10) and employing (4.22) and (4.61). When attempting to solve the Dubrovin systems (4.68) and (4.70), they must be augmented with appropriate divisors Dμˆ (n0 ,t0,r ) ∈ Symp Kp , t0,r ∈ Iμ , and Dνˆ(n0 ,t0,r ) ∈ Symp Kp , t0,r ∈ Iν , as initial conditions. Since the stationary trace formulas for fℓ,± and hℓ,± in terms of symmetric functions of the zeros µj and νk of ( · )p− Fp and ( · )p− −1 Hp in Lemma 3.2 extend line by line to the corresponding time-dependent setting, we next record their tr dependent analogs without proof. For simplicity we again confine ourselves to the simplest cases only. Lemma 4.5. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. Then, 2p+1

−1/2 p α , (4.73) µj = (−1)p+1 Em + α m=0 j=1

−1/2 2p+1 p β+ = (−1)p+1 , Em νj β m=0 j=1

p

j=1 p j=1

µj = α+ β − γ + νj = α+ β − γ

α++ c1,+ − , α+ c0,+

c1,+ β− − . β c0,+

(4.74)

(4.75)

(4.76)

Next, we turn to the asymptotic expansions of φ and Ψ in a neighborhood of P∞± and P0,± . Lemma 4.6. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. Moreover, let P = (z, y) ∈ Kp \ {P∞+ , P∞− , P0,+ , P0,− }, (n, n0 , tr , t0,r ) ∈ Z2 × R2 . Then φ has the asymptotic behavior / P → P∞+ , β + β − γζ + O(ζ 2 ), φ(P ) = ζ = 1/z, ζ→0 −(α+ )−1 ζ −1 + (α+ )−2 α++ γ + + O(ζ), P → P∞− , (4.77)

172


φ(P ) =

ζ→0

/

α−1 − α−2 α− γζ + O(ζ 2 ), −β + ζ − β ++ γ + ζ 2 + O(ζ 3 ),

P → P0,+ , P → P0,− ,

ζ = z.

(4.78)

The component ψ1 of the Baker–Akhiezer vector Ψ has the asymptotic behavior

r+ i ψ1 (P, n, n0 , tr , t0,r ) = exp ± (tr − t0,r ) c˜r+ −s,+ ζ −s (1 + O(ζ)) ζ→0 2 s=0 ⎧ n −n 0 ζ , P → P∞+ , ⎪ ⎪ ⎨ α+ (n,tr ) × ζ = 1/z, Γ(n, n0 , tr ) α+ (n0 ,t0,r ) ⎪ . P → P∞− , ⎪ ⎩× exp i tr dsg˜ (n , s) − g ˜ (n , s) , r ,+ 0 r ,− 0 + − t0,r

(4.79)

ψ1 (P, n, n0 , tr , t0,r ) = exp ζ→0

×

⎧ α(n,t ) r ⎪ , ⎪ ⎨ α(n0 ,t0,r )

r−

i ± (tr − t0,r ) c˜r− −s,− ζ −s (1 + O(ζ)) 2 s=0 n−n0

Γ(n, n0 , tr )ζ ⎪ . ⎪ ⎩× exp i tr dsg˜ ˜r− ,− (n0 , s) , r+ ,+ (n0 , s) − g t0,r

P → P0,+ , P → P0,− ,

ζ = z. (4.80)

Proof. Since by the definition of φ in (4.23) the time parameter tr can be viewed as an additional but fixed parameter, the asymptotic behavior of φ remains the same as in Lemma 3.3. Similarly, also the asymptotic behavior of ψ1 (P, n, n0 , tr , tr ) is derived in an identical fashion to that in Lemma 3.3. This proves (4.79) and (4.80) for t0,r = tr , that is, / ζ n0 −n (1 + O(ζ)), P → P∞+ , ψ1 (P, n, n0 , tr , tr ) = ζ = 1/z, α+ (n,t ) ζ→0 Γ(n, n0 , tr ) α+ (n0 ,trr ) + O(ζ), P → P∞− , (4.81) / α(n,t ) r + O(ζ), P → P , 0,+ α(n0 ,tr ) ψ1 (P, n, n0 , tr , tr ) = ζ = z. ζ→0 Γ(n, n0 , tr )ζ n−n0 (1 + O(ζ)), P → P0,− , (4.82) It remains to investigate

ψ1 (P, n0 , n0 , tr , t0,r ) = exp i

tr

t0,r

)r (z, n0 , t) − F)r (z, n0 , t)φ(P, n0 , t) . dt G

(4.83) The asymptotic expansion of the integrand is derived using Theorem A.2. Focusing on the homogeneous coefficients first, one computes as P → P∞± , * s,+ − F*s,+ *s,+ − F*s,+ φ = G G

Gp + (c0,+ /2)z −p− y Fp


−1

p− p− Gp 2z Fp *s,+ − F*s,+ 2z +1 =G c0,+ y c0,+ y

gˆ0,+ ∓ 21 ˆ 1 = ± ζ −s + fs,+ + O(ζ), ζ→0 2 fˆ0,+

P → P∞± , ζ = 1/z.

173

(4.84)

Since F)r =

ζ→0

r+ s=0

c˜r+ −s,+ F*s,+ + O(ζ),

one infers from (4.77)

)r = G

ζ→0

r+ s=0

* s,+ + O(ζ), c˜r+ −s,+ G

(4.85)

r+

) r − F)r φ = 1 c˜r+ −s,+ ζ −s + O(ζ), G ζ→0 2 s=0

P → P∞+ , ζ = 1/z.

(4.86)

Insertion of (4.86) into (4.83) then proves (4.79) as P → P∞+ . As P → P∞− , we need one additional term in the asymptotic expansion of F)r , that is, we will use F)r =

ζ→0

This then yields

) r − F)r φ = − 1 G ζ→0 2

r+ s=0

c˜r+ −s,+ F*s,+ +

r+ s=0

r− s=0

c˜r− −s,− fˆs−1,− ζ + O(ζ 2 ).

c˜r+ −s,+ ζ −s − (α+ )−1 (f˜r+ ,+ − f˜r− −1,− ) + O(ζ).

(4.87)

(4.88)

Invoking (2.34) and (4.2) one concludes that + gr+ ,+ − g˜r− ,− ) f˜r− −1,− − f˜r+ ,+ = −iα+ tr + α (˜

(4.89)

and hence ) r − F)r φ = − 1 G ζ→0 2

r+ s=0

c˜r+ −s,+ ζ −s −

iα+ tr α+

+ g˜r+ ,+ − g˜r− ,− + O(ζ).

(4.90)

Insertion of (4.90) into (4.83) then proves (4.79) as P → P∞− . Using Theorem A.2 again, one obtains in the same manner as P → P0,± ,

Since

1 *s,− − F*s,− φ = ± 1 ζ −s − gˆs,− + gˆ0,− ± 2 fˆs,− + O(ζ). G ˆ ζ→0 2 f0,−

F)r =

ζ→0

)r = G

ζ→0

r− s=0 r−

s=0

c˜r− −s,− F*s,− + f˜r+ −1,+ + O(ζ), * s,− + g˜r+ ,+ + O(ζ), c˜r− −s,− G

P → P0,± , ζ = z, P → P0,± , ζ = z,

(4.91)

(4.92) (4.93)

174


(4.91)–(4.93) yield ) r −F)r φ = ± 1 G ζ→0 2

r− s=0

c˜r− −s,− ζ −s +˜ gr+ ,+ −˜ gr− ,− −

gˆ0,− ± 21 ˜ (fr+ −1,+ −f˜r− ,− )+O(ζ), fˆ0,−

(4.94) where we again used (4.78), (2.52), and (4.2). As P → P0,− , one thus obtains )r − F)r φ = − 1 G ζ→0 2

r− s=0

c˜r− −s,− ζ −s + g˜r+ ,+ − g˜r− ,− ,

P → P0,− , ζ = z.

(4.95)

Insertion of (4.95) into (4.83) then proves (4.80) as P → P0,− . As P → P0,+ , one obtains r−

1 ) r − F)r φ = 1 G c˜r− −s,− ζ −s + g˜r+ ,+ − g˜r− ,− − (f˜r+ −1,+ − f˜r− ,− ) + O(ζ), ζ→0 2 α s=0 r

=

ζ→0

− iαtr 1 + O(ζ), c˜r −s,− ζ −s − 2 s=0 − α

P → P0,+ , ζ = z,

(4.96)

using f˜r− ,− = f˜r−− −1,− + α(˜ gr− ,− − g˜r−−,− ) (cf. (2.38)) and (4.2). Insertion of (4.96) into (4.83) then proves (4.80) as P → P0,+ . Next, we note that Lemma 3.4 on nonspecial divisors in the stationary context extends to the present time-dependent situation without a change. Indeed, since tr ∈ R just plays the role of a parameter, the proof of Lemma 3.4 extends line by line and is hence omitted. Lemma 4.7. Assume Hypothesis 4.1 and suppose that (4.7), (4.8) hold. Moreover, let (n, tr ) ∈ Z × R. Denote by Dμˆ , µ ˆ = {ˆ µ1 , . . . , µ ˆp } and Dνˆ , νˆ = {ˆ ν1 , . . . , νˆp }, the pole and zero divisors of degree p, respectively, associated with α, β, and φ defined according to (4.21) and (4.22), that is, µ ˆj (n, tr ) = (µj (n, tr ), (2/c0,+ )µj (n, tr )p− Gp (µj (n, tr ), n, tr )),

j = 1, . . . , p, (4.97)

νˆj (n, tr ) = (νj (n, tr ), −(2/c0,+ )νj (n, tr )p− Gp (νj (n, tr ), n, tr )),

j = 1, . . . , p. (4.98)

Then Dμˆ (n,tr ) and Dνˆ(n,tr ) are nonspecial for all (n, tr ) ∈ Z × R. Finally, we note that Γ(n, n0 , tr ) = Γ(n, n0 , t0,r )

tr ds g˜r+ ,+ (n, s) − g˜r+ ,+ (n0 , s) − g˜r− ,− (n, s) + g˜r− ,− (n0 , s) , × exp i t0,r

(4.99)


175

which follows from (2.91), (3.31), and from Γ(n, n0 , tr )tr =

n

γ(j, tr )tr

j=n0 +1

n

γ(j, tr )

(4.100)

k=n0 +1 k =j

= i g˜r+ ,+ (n, tr ) − g˜r+ ,+ (n0 , tr ) − g˜r− ,− (n, tr ) + g˜r− ,− (n0 , tr ) Γ(n, n0 , tr )

after integration with respect to tr . The results of Sections 2–4 have been used extensively in [31] to derive the class of time-dependent algebro-geometric solutions of the Ablowitz–Ladik hierarchy and the associated theta function representations of α, β, φ, and Ψ. These theta / {0, 1} for all (n, tr ) ∈ Z × R, function representations also show that γ(n, tr ) ∈ and hence condition (4.1) is satisfied for the time-dependent algebro-geometric AL solutions discussed in this section, provided the associated divisors Dμˆ (n,tr ) and Dνˆ(n,tr ) stay away from P∞± , P0,± for all (n, tr ) ∈ Z × R.

Appendix A. Asymptotic spectral parameter expansions and nonlinear recursion relations In this appendix we consider asymptotic spectral parameter expansions of Fp /y, Gp /y, and Hp /y in a neighborhood of P∞± and P0,± , the resulting recursion ˆ ℓ , their connection with the relations for the homogeneous coefficients fˆℓ , gˆℓ , and h nonhomogeneous coefficients fℓ , gℓ , and hℓ , and the connection between cℓ,± and cℓ (E ±1 ). We will employ the notation ±1 E ±1 = E0±1 , . . . , E2p+1 . (A.1)

We start with the following elementary results (consequences of the binomial expansion) assuming η ∈ C such that |η| < min{|E0 |−1 , . . . , |E2p+1 |−1 }: 02p+1 1−1/2 ∞ (A.2) = cˆk (E)η k , 1 − Em η m=0

k=0

where

cˆ0 (E) = 1, cˆk (E) =

k

(2j0 )! · · · (2j2p+1 )! j2p+1 E j0 · · · E2p+1 , 22k (j0 !)2 · · · (j2p+1 !)2 0

j0 ,...,j2p+1 =0 j0 +···+j2p+1 =k

k ∈ N.

(A.3)

The first few coefficients explicitly read cˆ0 (E) = 1, cˆ1 (E) =

2p+1 2p+1 2p+1 1 1 3 2 Em , cˆ2 (E) = E , Em1 Em2 + 2 m=0 4 m ,m =0 8 m=0 m

etc.

1 2 m1 <m2

(A.4)

176


Similarly,

02p+1 m=0

where

1 − Em η

11/2

=

∞

ck (E)η k ,

(A.5)

k=0

c0 (E) = 1, ck (E) =

j

2p+1 (2j0 )! · · · (2j2p+1 )! E0j0 · · · E2p+1 , 2 · · · (j2p+1 !) (2j0 − 1) · · · (2j2p+1 − 1)

k

22k (j0 !)2 j0 ,...,j2p+1 =0 j0 +···+j2p+1 =k

k ∈ N. (A.6)

The first few coefficients explicitly are given by c0 (E) = 1, c1 (E) = −

2p+1 2p+1 2p+1 1 1 1 2 Em , c2 (E) = E , Em1 Em2 − 2 m=0 4 m ,m =0 8 m=0 m 1

etc.

2

m1 <m2

(A.7) k

Multiplying (A.2) and (A.5) and comparing coefficients of η one finds k

cˆk−ℓ (E)cℓ (E) = δk,0 ,

ℓ=0

k ∈ N0 .

(A.8)

Next, we turn to asymptotic expansions of various quantities in the case of / {0, 1}, n ∈ Z. the Ablowitz–Ladik hierarchy assuming α, β ∈ CZ , α(n)β(n) ∈ Consider a fundamental system of solutions Ψ± (z, · ) = (ψ1,± (z, · ), ψ2,± (z, · ))⊤ of U (z)Ψ− ± (z) = Ψ± (z) for z ∈ C (or in some subdomain of C), with U given by (2.5), such that (A.9) det(Ψ− (z), Ψ+ (z)) = 0. Introducing

φ± (z, n) =

ψ2,± (z, n) , ψ1,± (z, n)

z ∈ C, n ∈ N,

(A.10)

then φ± satisfy the Riccati-type equation − αφ± φ− ± − φ± + zφ± = zβ,

(A.11)

and one introduces in addition, 2 , φ+ − φ− φ+ + φ− , g= φ+ − φ− 2φ+ φ− h= . φ+ − φ− f=

(A.12) (A.13) (A.14)


177

Using the Riccati-type equation (A.11) and its consequences, − − − α(φ+ φ− + − φ− φ− ) − (φ+ − φ− ) + z(φ+ − φ− ) = 0,

α(φ+ φ− +

+

φ− φ− −)

−

(φ− +

+

φ− −)

+ z(φ+ + φ− ) = 2zβ,

(A.15) (A.16)

one derives the identities z(g− − g) + zβf + αh− = 0, −

(A.17)

−

zβf + αh − g + g = 0,

(A.18)

−f + zf− + α(g + g− ) = 0,

(A.19)

g2 − fh = 1.

(A.21)

zβ(g− + g) − zh + h− = 0,

(A.20)

Moreover, (A.17)–(A.20) and (A.21) also permit one to derive nonlinear difference equations for f, g, and h separately, and one obtains + 2 (α + zα)2 f − z(α+ )2 γf− − 2zα2 γ + (α+ + zα)2 f + z(α+ )2 γf− f+ + z 2 α4 (γ + )2 (f+ )2 = 4(αα+ )2 (α+ + αz)2 ,

+

+

+

+

(α + zα)(β + zβ )(z + α β)(1 + zαβ )g

(A.22)

2

+ z(α+ γg− + zαγ + g+ )(zβ + γg− + βγ + g+ ) − zγ (α+ β + z 2 αβ + )(2 − γ + ) + 2z(1 − γ + )(2 − γ) g− g − zγ + 2z(1 − γ)(2 − γ + ) + (α+ β + z 2 αβ + )(2 − γ) g+ g

(A.23)

+ (β + zβ + )4 h2 = 4z 2 (ββ + )2 (β + β + z)2 .

(A.24)

= (α+ β − z 2 αβ + )2 , 2 z 2 (β + )2 γh− − β 2 γ + h+ − 2z(β + zβ + )2 (β + )2 γh− + β 2 γ + h+ h

For the precise connection between f, g, h and the Green’s function of the Lax difference expression underlying the AL hierarchy, we refer to [30, App. C], [33]. Next, we assume the existence of the following asymptotic expansions of f, g, and h near 1/z = 0 and z = 0. More precisely, near 1/z = 0 we assume that f(z)

h(z)

=

|z|→∞ z∈CR

=

|z|→∞ z∈CR

−

∞

−

ˆfℓ,+ z −ℓ−1 ,

ℓ=0

∞

g(z)

=

|z|→∞ z∈CR

−

∞ ℓ=0

ˆgℓ,+ z −ℓ , (A.25)

ˆℓ,+ z −ℓ , h

ℓ=0

for z in some cone CR with apex at z = 0 and some opening angle in (0, 2π], exterior to a disk centered at z = 0 of sufficiently large radius R > 0, for some set

178


ˆℓ,+ , ℓ ∈ N0 . Similarly, near z = 0 we assume that of coefficients ˆfℓ,+ , ˆgℓ,+ , and h ∞

f(z) = − |z|→0 z∈Cr

ˆfℓ,− z ℓ ,

ℓ=0

∞

h(z) = − |z|→0 z∈Cr

g(z) = − |z|→0 z∈Cr

∞

ˆgℓ,− z ℓ ,

ℓ=0

(A.26)

ˆℓ,− z ℓ+1 , h

ℓ=0

for z in some cone Cr with apex at z = 0 and some opening angle in (0, 2π], interior to a disk centered at z = 0 of sufficiently small radius r > 0, for some set ˆℓ,− , ℓ ∈ N0 . Then one can prove the following result. of coefficients ˆfℓ,− , ˆgℓ,− , and h Theorem A.1. Assume α, β ∈ CZ , α(n)β(n) ∈ / {0, 1}, n ∈ Z, and the existence of the asymptotic expansions (A.25) and (A.26). Then f, g, and h have the following asymptotic expansions as |z| → ∞, z ∈ CR , respectively, |z| → 0, z ∈ Cr , f(z)

=

|z|→∞ z∈CR

h(z)

=

|z|→∞ z∈CR

−

∞

fˆℓ,+ z −ℓ−1 ,

g(z)

ℓ=0

−

∞

=

|z|→∞ z∈CR

−

∞

gˆℓ,+ z −ℓ ,

ℓ=0

(A.27)

ˆ ℓ,+ z −ℓ , h

ℓ=0

and f(z) = − |z|→0 z∈Cr

∞

h(z) = − |z|→0 z∈Cr

fˆℓ,− z ℓ ,

ℓ=0

∞

g(z) = − |z|→0 z∈Cr

∞ ℓ=0

gˆℓ,− z ℓ , (A.28)

ˆ ℓ,− z ℓ+1 , h

ℓ=0

where fˆℓ,± , gˆℓ,± , and ˆhℓ,± are the homogeneous versions of the coefficients fℓ,± , ˆ ℓ,± can be gℓ,± , and hℓ,± defined in (2.49)–(2.51). In particular, fˆℓ,± , gˆℓ,± , and h computed from the following nonlinear recursion relations3 fˆ1,+ = (α+ )2 β − γ + α++ , + 3 2 ˆ f2,+ = −(α ) β + γ(α+ )2 β − + γ + (α++ )2 β + − γ ++ α+++ + 2α+ α++ β , ℓ−1 ℓ−4 1 (α+ )4 fˆm,+ fˆℓ−m,+ fˆm,+ fˆℓ−m−4,+ + α4 α4 α+ fˆℓ,+ = 2 m=1 m=0

fˆ0,+ = −α+ ,

− 2(α+ )2

ℓ−3

m=0

− + + α2 γ + fˆℓ−m−3,+ fˆm,+ − 2αα+ fˆℓ−m−3,+ + (α+ )2 γ fˆℓ−m−3,+

3 We recall, a sum is interpreted as zero whenever the upper limit in the sum is strictly less than its lower limit.


+

ℓ−2

179

+ + − − fˆℓ−m−2,+ + (α+ )2 γ fˆm,+ ((α+ )2 γ fˆℓ−m−2,+ α4 (γ + )2 fˆm,+

m=0

+ − 2α2 γ + fˆℓ−m−2,+ )

− + + 2α2 γ + fˆℓ−m−2,+ ) − 2αα+ fˆm,+ (−3αα+ fˆℓ−m−2,+ + 2(α+ )2 γ fˆℓ−m−2,+

ℓ−1 − + − 2α2 fˆm,+ − 2αα+ fˆℓ−m−1,+ + (α+ )2 γ fˆℓ−m−1,+ , + α2 γ + fˆℓ−m−1,+ m=0

ℓ ≥ 3, (A.29)

fˆ1,− = γα− − α2 β + , 3 + 2 ˆ f2,− = α (β ) − γ + α2 β ++ − γ (α− )2 β − γ − α−− + 2α− αβ + , ℓ−4 ℓ−1 1 fˆm,− fˆℓ−m−4,− + (α+ )4 fˆm,− fˆℓ−m,− α(α+ )4 fˆℓ,− = − α4 2 m=0 m=1

fˆ0,− = α,

− 2α2 +

ℓ−3

m=0

ℓ−2

− + + α2 γ + fˆℓ−m−3,− fˆm,− − 2αα+ fˆℓ−m−3,− + (α+ )2 γ fˆℓ−m−3,−

+ + α4 (γ + )2 fˆm,− fˆℓ−m−2,−

m=0

− − + + (α+ )2 γ fˆm,− ((α+ )2 γ fˆℓ−m−2,− − 2α2 γ + fˆℓ−m−2,− )

− + + 2α2 γ + fˆℓ−m−2,− ) − 2αα+ fˆm,− (−3αα+ fˆℓ−m−2,− + 2(α+ )2 γ fˆℓ−m−2,−

− 2(α+ )2

ℓ−1

m=0

− + , + α2 γ + fˆℓ−m−1,− fˆm,− − 2αα+ fˆℓ−m−1,− + (α+ )2 γ fˆℓ−m−1,− ℓ ≥ 3, (A.30)

1 , gˆ1,+ = −α+ β, 2 gˆ2,+ = (α+ β)2 − γ + α++ β − γα+ β + , ℓ−4 ℓ−1 gˆm,+ gˆℓ−m−4,+ + α2 (β + )2 gˆm,+ gˆℓ−m,+ (αβ + )2 gˆℓ,+ = − (α+ )2 β 2 gˆ0,+ =

m=0

+ α+ β

ℓ−3

m=0

+

ℓ−2

m=0

m=1

− + γγ + gˆm,+ gˆℓ−m−3,+ + gˆm,+ ((1 + αβ)(1 + α+ β + )ˆ gℓ−m−3,+

− + − (γ + α+ β + γ)ˆ gℓ−m−3,+ + (−2 + γ)γ + gˆℓ−m−3,+ )

− − + + gˆℓ−m−2,+ + αβ(γ + )2 gˆm,+ gˆℓ−m−2,+ α+ β + γ 2 gˆm,+

180

F. Gesztesy, H. Holden, J. Michor and G. Teschl + gˆm,+ ((α+ β + + α2 α+ β 2 β + + αβ(1 + α+ β + )2 )ˆ gℓ−m−2,+

+ αβ +

− + − 2(α+ (1 + αβ)β + γˆ gℓ−m−2,+ + αβ(1 + α+ β + )γ + gˆℓ−m−2,+ )

ℓ−1

− + γγ + gˆm,+ gˆℓ−m−1,+ + gˆm,+ ((1 + αβ)(1 + α+ β + )ˆ gℓ−m−1,+

m=0

− + − (γ + α+ β + γ)ˆ gℓ−m−1,+ + (−2 + γ)γ + gˆℓ−m−1,+ ) ,

ℓ ≥ 3, (A.31)

1 , gˆ1,− = −αβ + , 2 gˆ2,− = (αβ + )2 − γ + αβ ++ − γα− β + , ℓ−4 ℓ−1 gˆm,− gˆℓ−m−4,− + (α+ )2 β 2 gˆm,− gˆℓ−m,− (α+ )2 β 2 gˆℓ,− = − α2 (β + )2 gˆ0,− =

m=0

+ αβ +

ℓ−3

− + + gˆm,− ((1 + αβ)(1 + α+ β + )ˆ gℓ−m−3,− gˆℓ−m−3,− γγ + gˆm,−

m=0

+

ℓ−2

m=1

− + − (γ + α+ β + γ)ˆ gℓ−m−3,− + (−2 + γ)γ + gˆℓ−m−3,− )

− − + + α+ β + γ 2 gˆm,− gˆℓ−m−2,− + αβ(γ + )2 gˆm,− gˆℓ−m−2,−

m=0

+ α+ β

+ gˆm,− ((α+ β + + α2 α+ β 2 β + + αβ(1 + α+ β + )2 )ˆ gℓ−m−2,−

− + − 2(α+ (1 + αβ)β + γˆ gℓ−m−2,− + αβ(1 + α+ β + )γ + gˆℓ−m−2,− ))

ℓ−1

− + γγ + gˆm,− gˆℓ−m−1,− + gˆm,− ((1 + αβ)(1 + α+ β + )ˆ gℓ−m−1,−

m=0

− + − (γ + α+ β + γ)ˆ gℓ−m−1,− + (−2 + γ)γ + gˆℓ−m−1,− ) ,

ˆ 1,+ = γβ − − α+ β 2 , h ˆ 2,+ = (α ) β − γ + α++ β 2 − γ α(β − )2 − γ − β −− + 2α+ β − β , h ℓ−4 ℓ−1 ˆ ℓ,+ = − 1 β 4 ˆ m,+ h ˆ ℓ−m−4,+ + (β + )4 ˆ m,+ h ˆ ℓ−m,+ β(β + )4 h h h 2 m=0 m=1

ℓ ≥ 3, (A.32)

ˆ 0,+ = β, h

+ 2 3

− 2β 2 +

ℓ−3

m=0

ℓ−2

m=0

2 +ˆ+ ˆ m,+ − 2ββ + h ˆ ℓ−m−3,+ + (β + )2 γ ˆh− h ℓ−m−3,+ + β γ hℓ−m−3,+

ˆ+ h ˆ+ β 4 (γ + )2 h m,+ ℓ−m−2,+


181

+ 2 ˆ− 2 +ˆ + + (β + )2 γ ˆh− m,+ ((β ) γ hℓ−m−2,+ − 2β γ hℓ−m−2,+ )

2 +ˆ+ ˆ m,+ (−3ββ + h ˆ ℓ−m−2,+ + 2(β + )2 γ ˆh− − 2ββ + h ℓ−m−2,+ + 2β γ hℓ−m−2,+ )

− 2(β + )2

ℓ−1

m=0

2 +ˆ + ˆhm,+ − 2ββ + h ˆ ℓ−m−1,+ + (β + )2 γ ˆh− ℓ−m−1,+ + β γ hℓ−m−1,+ , ℓ ≥ 3, (A.33)

ˆ 0,− = −β + , h

ˆ 1,− = −γ + β ++ + α(β + )2 , h 2 + 3 ˆ h2,− = −α (β ) + γα− (β + )2 + γ α+ (β ++ )2 − γ ++ β +++ + 2αβ + β ++ , ℓ−4 ℓ−1 ˆ ℓ,− = 1 (β + )4 ˆ m,− h ˆ ℓ−m−4,− + β 4 ˆ m,− h ˆ ℓ−m,− h h β+β4h 2 m=0 m=1 − 2(β + )2 +

ℓ−2

ℓ−3

m=0

2 + ˆ+ ˆhm,− − 2ββ + h ˆ ℓ−m−3,− + (β + )2 γ ˆh− hℓ−m−3,− ℓ−m−3,− + β γ

ˆ + ˆh+ β 4 (γ + )2 h m,− ℓ−m−2,−

m=0

+ 2 ˆ− 2 +ˆ+ + (β + )2 γ ˆh− m,− ((β ) γ hℓ−m−2,− − 2β γ hℓ−m−2,− )

2 +ˆ+ ˆ m,− (−3ββ + h ˆ ℓ−m−2,− + 2(β + )2 γ ˆh− − 2ββ + h ℓ−m−2,− + 2β γ hℓ−m−2,− )

ℓ−1 2 +ˆ+ ˆ m,− − 2ββ + h ˆ ℓ−m−1,− + (β + )2 γ ˆh− − 2β 2 h , ℓ−m−1,− + β γ hℓ−m−1,− m=0

ℓ ≥ 3. (A.34)

Proof. We first consider the expansions (A.27) near 1/z = 0 and the nonlinear recursion relations (A.29), (A.31), and (A.33) in detail. Inserting expansion (A.25) for f into (A.22), the expansion (A.25) for g into (A.23), and the expansion (A.25) for h into (A.24), then yields the nonlinear recursion relations (A.29), (A.31), and ˆℓ,+ , respectively. (A.33), but with fˆℓ,+ , gˆℓ,+ , and ˆhℓ,+ replaced by ˆfℓ,+ , ˆgℓ,+ , and h From the leading asymptotic behavior one finds that ˆf0,+ = −α+ , ˆg0,+ = 21 , and ˆ0,+ = β. h Next, inserting the expansions (A.25) for f, g, and h into (A.17)–(A.20), and comparing powers of z −ℓ as |z| → ∞, z ∈ CR , one infers that fℓ,+ , gℓ,+ , and hℓ,+ satisfy the linear recursion relations (2.32)–(2.35). Here we have used (2.21). The ˆ0,+ are consistent with (2.32) for c0,+ = 1. Hence one coefficients ˆf0,+ , ˆg0,+ , and h concludes that ˆfℓ,+ = fℓ,+ , ˆgℓ,+ = gℓ,+ , h ˆℓ,+ = hℓ,+ , ℓ ∈ N0 , (A.35)

for certain values of the summation constants cℓ,+ . To conclude that actually, ˆfℓ,+ = fˆℓ,+ , ˆgℓ,+ = gˆℓ,+ , h ˆℓ,+ = h ˆ ℓ,+ , ℓ ∈ N0 , and hence all cℓ,+ , ℓ ∈ N, vanish, we now rely on the notion of degree as introduced in Remark 2.6. To this end we

182


recall that deg fˆℓ,+ = ℓ + 1,

deg gˆℓ,+ = ℓ,

ˆ ℓ,+ = ℓ, deg h

ℓ ∈ N0 ,

(A.36)

(cf. (2.55)). Similarly, the nonlinear recursion relations (A.29), (A.31), and (A.33) yield inductively that ˆℓ,+ = ℓ, ℓ ∈ N0 . (A.37) deg ˆfℓ,+ = ℓ + 1, deg ˆgℓ,+ = ℓ, deg h Hence one concludes ˆfℓ,+ = fˆℓ,+ ,

ˆgℓ,+ = gˆℓ,+ ,

ˆℓ,+ = h ˆ ℓ,+ , h

ℓ ∈ N0 .

(A.38)

The proof of the corresponding asymptotic expansion (A.28) and the nonlinear recursion relations (A.30), (A.32), and (A.34) follows precisely the same strategy and is hence omitted. Given this general result on asymptotic expansions, we now specialize to the algebro-geometric case at hand. We recall our conventions y(P ) = ∓(ζ −p−1 + O(ζ −p )) for P near P∞± (where ζ = 1/z) and y(P ) = ±((c0,− /c0,+ ) + O(ζ)) for P near P0,± (where ζ = z). Theorem A.2. Assume (3.1), s-ALp (α, β) = 0, and suppose P = (z, y) {P∞+ , P∞− }. Then z p− Fp /y, z p− Gp /y, and z p− Hp /y have the following gent expansions as P → P∞± , respectively, P → P0,± , / ∞ z p− Fp (z) ζ = 1/z, ∓ ℓ=0 fˆℓ,+ ζ ℓ+1 , P → P∞± , = ∞ ˆ c0,+ y ± ℓ=0 fℓ,− ζ ℓ , P → P0,± , ζ = z, / ∞ ζ = 1/z, ∓ ℓ=0 gˆℓ,+ ζ ℓ , P → P∞± , z p− Gp (z) ∞ = c0,+ y ± ℓ=0 gˆℓ,− ζ ℓ , P → P0,± , ζ = z, / ℓ ˆ P → P∞± , ζ = 1/z, ∓ ∞ z p− Hp (z) ℓ=0 hℓ,+ ζ , = ∞ ˆ c0,+ y ± ℓ=0 hℓ,− ζ ℓ+1 , P → P0,± , ζ = z,

∈ Kp \ conver-

(A.39) (A.40) (A.41)

where ζ = 1/z (resp., ζ = z) is the local coordinate near P∞± (resp., P0,± ) and ˆ ℓ,± are the homogeneous versions4 of the coefficients fℓ,± , gℓ,± , fˆℓ,± , gˆℓ,± , and h and hℓ,± as introduced in (2.49)–(2.51). Moreover, one infers for the Em -dependent summation constants cℓ,± , ℓ = 0, . . . , p± , in Fp , Gp , and Hp that cℓ,± = c0,± cℓ E ±1 , ℓ = 0, . . . , p± . (A.42) In addition, one has the following relations between the homogeneous and nonhomogeneous recursion coefficients: fℓ,± = c0,±

ℓ

k=0

cℓ−k E ±1 fˆk,± ,

ℓ = 0, . . . , p± ,

(A.43)

4 Strictly speaking, the coefficients fˆ ˆ ℓ,± in (A.39)–(A.41) no longer have a wellˆℓ,± , and h ℓ,± , g defined degree and hence represent a slight abuse of notation since we assumed that s-AL p (α, β) = 0. At any rate, they are explicitly given by (A.49)–(A.51).


gℓ,± = c0,±

ℓ

k=0

hℓ,± = c0,±

ℓ

k=0

cℓ−k E ±1 gˆk,± ,

183

ℓ = 0, . . . , p± ,

(A.44)

ℓ = 0, . . . , p± .

(A.45)

ℓ = 0, . . . , p± − 1,

(A.46)

cℓ−k E ±1 hk,± ,

Furthermore, one has c0,± fˆℓ,± =

ℓ

k=0

c0,± fˆp± ,± =

cˆℓ−k E ±1 fk,± ,

p± −1

k=0

c0,± gˆℓ,± =

c0,± gˆp± ,± =

ℓ

cˆℓ−k E ±1 gk,± ,

k=0 p± −1

k=0

ˆ ℓ,± = c0,± h ˆ p ,± = c0,± h ±

ℓ

cˆp± −k E ±1 fk,± + cˆ0 (E ±1 )fp∓ −1,∓ ,

k=0

ℓ = 0, . . . , p± − 1,

ˆ ℓ,± c0,± h

5m

⎧ ℓ ⎪ ˆℓ−k E ±1 gk,± , ⎨ k=0 c ±1 p± −δ± gk,± cˆℓ−k E = k=0 ±1 ⎪ ⎩ + p∓ −δ± gk,∓ , c ˆ E ℓ+k−p k=(p−ℓ)∨0

⎧ ℓ ⎪ ˆℓ−k E ±1 hk,± , ⎨ k=0 c p± −1 = ˆℓ−k E ±1 hk,± k=0 c ⎪ ⎩ + p∓ −1 ˆℓ+k−p E ±1 hk,∓ , k=(p−ℓ)∨0 c

∨ n = max{m, n}.

(A.48)

cˆp± −k E ±1 hk,± + cˆ0 (E ±1 )hp∓ −1,∓ .

For general ℓ (not restricted to ℓ ≤ p± ) one has5 ⎧ ℓ ⎪ ˆℓ−k E ±1 fk,± , ⎨ k=0 c p± −1 ˆℓ−k E ±1 fk,± c0,± fˆℓ,± = k=0 c ⎪ ⎩ + p∓ −1 ˆℓ+k−p E ±1 fk,∓ , k=(p−ℓ)∨0 c c0,± gˆℓ,±

(A.47)

cˆp± −k E ±1 gk,± + cˆ0 (E ±1 )gp∓ ,∓ ,

cˆℓ−k E ±1 hk,± ,

k=0 p± −1

ℓ = 0, . . . , p± − 1,

ℓ = 0, . . . , p± − 1, ℓ ≥ p± , ℓ = 0, . . . , p± − δ± , ℓ ≥ p± − δ± + 1, ℓ = 0, . . . , p± − 1, ℓ ≥ p± .

(A.49)

(A.50)

(A.51)

184


Here we used the convention

/

δ± =

0, 1,

+, −.

(A.52)

Proof. Identifying Ψ+ (z, · ) with Ψ(P, · , 0) and Ψ− (z, · ) with Ψ(P ∗ , · , 0), ∗

recalling that W (Ψ(P, · , 0), Ψ(P , · , 0)) = −c0,+ z (3.30)), and similarly, identifying

n−n0 −p−

(A.53) −1

yFp (z, 0)

Γ(n, n0 ) (cf.

φ+ (z, · ) with φ(P, · ) and φ− (z, · ) with φ(P ∗ , · ),

(A.54)

a comparison of (A.10)–(A.14) and the results of Lemmas 3.1 and 3.3 shows that we may also identify f with ∓

2Fp c0,+ z −p− y

g with ∓

,

and h with ∓

2Gp c0,+ z −p− y

2Hp c0,+ z −p− y

, (A.55)

,

the sign depending on whether P tends to P∞± or to P0,± . In particular, (A.17)– (A.24) then correspond to (2.10)–(2.13), (2.69), (2.76)–(2.78), respectively. Since z p− Fp /y, z p− Gp /y, and z p− Hp /y clearly have asymptotic (in fact, even convergent) expansions as |z| → ∞ and as |z| → 0, the results of Theorem A.1 apply. Thus, as P → P∞± , one obtains the following expansions using (A.2) and (2.18)– (2.20):

p+ −1 p− ∞ z p− Fp (z) 1 cˆk (E)ζ k fp+ −1−ℓ,+ ζ p+ −ℓ , fp− −ℓ,− ζ p+ +ℓ + = ∓ c0,+ y ζ→0 c0,+ k=0

= ∓

ζ→0

∞

fˆℓ,+ ζ

ℓ+1

ℓ=1

ℓ=0

,

(A.56)

ℓ=0

p− p+ ∞ z p− Gp (z) 1 = ∓ gp− −ℓ,− ζ p+ +ℓ + gp+ −ℓ,+ ζ p+ −ℓ cˆk (E)ζ k c0,+ y ζ→0 c0,+ k=0

= ∓

ζ→0

∞

ℓ=1

ℓ=0

gˆℓ,+ ζ ℓ ,

(A.57)

ℓ=0

p p+ − −1 ∞ z p− Hp (z) 1 = ∓ hp+ −ℓ,+ ζ p+ −ℓ hp− −1−ℓ,− ζ p+ +ℓ + cˆk (E)ζ k c0,+ y ζ→0 c0,+ k=0

= ∓

ζ→0

∞

ℓ=0

ˆ ℓ,+ ζ . h ℓ

ℓ=0

This implies (A.39)–(A.41) as P → P∞± .

ℓ=1

(A.58)


185

Similarly, as P → P0,± , (A.2) and (2.18)–(2.20), and (2.74) imply z p− Fp (z) 1 = ± c0,+ y ζ→0 c0,−

= ±

ζ→0

cˆk (E −1 )ζ k

k=0

p−

× ∞

∞

ℓ=1

fp− −ℓ,− ζ p+ −ℓ +

p+ −1

ℓ=0

fp+ −1−ℓ,+ ζ p+ +ℓ

fˆℓ,− ζ , ℓ

(A.59)

ℓ=0

∞ z p− Gp (z) 1 = ± cˆk (E −1 )ζ k c0,+ y ζ→0 c0,− k=0

p+ p− × gp+ −ℓ,+ ζ p+ +ℓ gp− −ℓ,− ζ p+ −ℓ + ℓ=0

ℓ=1

= ±

ζ→0

∞

ℓ

gˆℓ,− ζ ,

(A.60)

ℓ=0

z p− Hp (z) 1 = ± c0,+ y ζ→0 c0,− × = ±

ζ→0

∞

ˆhℓ,− ζ

∞

k=0

p − −1 ℓ=0

ℓ+1

cˆk (E −1 )ζ k

hp− −1−ℓ,− ζ p+ −ℓ +

p+ ℓ=1

hp+ −ℓ,+ ζ p+ +ℓ

.

(A.61)

ℓ=0

Thus, (A.39)–(A.41) hold as P → P0,± . Next, comparing powers of ζ in the second and third term of (A.56), formula (A.46) follows (and hence (A.49) as well). Formulas (A.47) and (A.48) follow by using (A.57) and (A.58), respectively. To prove (A.43) one uses (A.8) and finds c0,±

ℓ

m=0

ℓ m cℓ−m (E) cℓ−m E ±1 fˆm,± = cˆm−k E ±1 fk,± m=0

= fℓ,± .

k=0

(A.62)

The proofs of (A.44) and (A.45) and those of (A.50) and (A.51) are analogous.

Finally, we also mention the following system of recursion relations for the ˆ ℓ,± . homogeneous coefficients fˆℓ,± , gˆℓ,± , and h

186


Lemma A.3. The homogeneous coefficients fˆℓ,± , gˆℓ,± , and ˆhℓ,± are uniquely defined by the following recursion relations: 1 ˆ 0,+ = β, gˆ0,+ = , fˆ0,+ = −α+ , h 2 l l fˆl−k,+ ˆhk,+ − gˆl+1,+ = gˆl+1−k,+ gˆk,+ , (A.63) k=0

k=1

− − = fˆl,+ − α(ˆ gl+1,+ + gˆl+1,+ ), fˆl+1,+

− ˆ l+1,+ = ˆh− + β(ˆ gl+1,+ + gˆl+1,+ ), h l,+

and 1 ˆ 0,− = −β + , , fˆ0,− = α, h 2 l l ˆ k,− − fˆl−k,− h = gˆl+1−k,− gˆk,− ,

gˆ0,− = gˆl+1,−

k=0

k=1

(A.64)

− − + α(ˆ gl+1,− + gˆl+1,− ), fˆl+1,− = fˆl,−

− ˆ ˆh− gl+1,− + gˆl+1,− ). l+1,− = hl,− − β(ˆ

Proof. One verifies that the coefficients defined via these recursion relations satisfy (2.32)–(2.35) (respectively, (2.36)–(2.39)). Since they are homogeneous of the required degree this completes the proof. Acknowledgments F.G., J.M., and G.T. gratefully acknowledge the extraordinary hospitality of the Department of Mathematical Sciences of the Norwegian University of Science and Technology, Trondheim, during extended stays in the summers of 2004–2006, where parts of this paper were written. F.G. and G.T. would like to thank all organizers of the international conference on Operator Theory and Mathematical Physics (OTAMP), Lund, June 2006, and especially, Pavel Kurasov, for their kind invitation and the stimulating atmosphere during the meeting. We are indebted to the anonymous referee for constructive remarks.

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Fritz Gesztesy Department of Mathematics, University of Missouri Columbia, MO 65211, USA e-mail: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html Helge Holden Department of Mathematical Sciences Norwegian University of Science and Technology NO–7491 Trondheim, Norway e-mail: [email protected] URL: http://www.math.ntnu.no/~holden/ Johanna Michor Faculty of Mathematics, University of Vienna Nordbergstrasse 15, 1090 Wien, Austria and International Erwin Schr¨ odinger Institute for Mathematical Physics Boltzmanngasse 9, 1090 Wien, Austria e-mail: [email protected] URL: http://www.mat.univie.ac.at/~jmichor/ Gerald Teschl Faculty of Mathematics, University of Vienna Nordbergstrasse 15, 1090 Wien, Austria and International Erwin Schr¨ odinger Institute for Mathematical Physics Boltzmanngasse 9, 1090 Wien, Austria e-mail: [email protected] URL: http://www.mat.univie.ac.at/~gerald/


On Dirichlet-to-Neumann Maps and Some Applications to Modified Fredholm Determinants Fritz Gesztesy, Marius Mitrea, and Maxim Zinchenko Dedicated with great pleasure to Boris Pavlov on the occasion of his 70th birthday

Abstract. We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schr¨ odinger operators in L2 (Ω; dn x), where Ω ⊂ Rn , n = 2, 3, are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in L2 (Ω; dn x) to modified Fredholm perturbation determinants associated with operators in L2 (∂Ω; dn−1 σ), n = 2, 3. This leads to a two- and three-dimensional extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schr¨ odinger operator on the half-line (0, ∞) to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation. Mathematics Subject Classification (2000). Primary: 47B10, 47G10, Secondary: 34B27, 34L40. Keywords. Fredholm determinants, non-self-adjoint operators, multi-dimensional Schr¨ odinger operators, Dirichlet-to-Neumann maps.

1. Introduction To describe the original Fredholm determinant result due to Jost and Pais [34], we D N need a few preparations. Denoting by H0,+ and H0,+ the one-dimensional Dirichlet and Neumann Laplacians in L2 ((0, ∞); dx), and assuming V ∈ L1 ((0, ∞); dx),

(1.1)

Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0405526, DMS-0400639, and FRG-0456306.

192

F. Gesztesy, M. Mitrea, and M. Zinchenko

D N we introduce the perturbed Schr¨ odinger operators H+ and H+ in L2 ((0, ∞); dx) by D H+ f = −f ′′ + V f, D f ∈ dom H+ = {g ∈ L2 ((0, ∞); dx) | g, g ′ ∈ AC([0, R]) for all R > 0,

(1.2)

N H+ f = −f ′′ + V f, N f ∈ dom H+ = {g ∈ L2 ((0, ∞); dx) | g, g ′ ∈ AC([0, R]) for all R > 0,

(1.3)

′′

2

g(0) = 0, (−g + V g) ∈ L ((0, ∞); dx)},

′

′′

2

g (0) = 0, (−g + V g) ∈ L ((0, ∞); dx)}.

D H+

N H+

Thus, and are self-adjoint if and only if V is real-valued, but since the latter restriction plays no special role in our results, we will not assume realvaluedness of V throughout this paper. D A fundamental system of solutions φD + (z, ·), θ+ (z, ·), and the Jost solution f+ (z, ·) of −ψ ′′ (z, x) + V ψ(z, x) = zψ(z, x),

z ∈ C\{0}, x ≥ 0,

(1.4)

are then introduced via the standard Volterra integral equations x −1/2 ′ sin(z 1/2 x) + dx′ z −1/2 sin(z 1/2 (x − x′ ))V (x′ )φD φD + (z, x) = z + (z, x ), 0

D θ+ (z, x)

1/2

x

′

(1.5)

−1/2

1/2

′

′

D )θ+ (z, x′ ),

dx z sin(z (x − x ))V (x x) + ∞ 0 1/2 f+ (z, x) = eiz x − dx′ z −1/2 sin(z 1/2 (x − x′ ))V (x′ )f+ (z, x′ ), = cos(z

(1.6) (1.7)

x

z ∈ C\{0}, Im(z 1/2 ) ≥ 0, x ≥ 0.

In addition, we introduce u = exp(i arg(V ))|V |1/2 ,

v = |V |1/2 , so that V = u v,

(1.8)

and denote by I+ the identity operator in L2 ((0, ∞); dx). Moreover, we denote by W (f, g)(x) = f (x)g ′ (x) − f ′ (x)g(x), 1

x ≥ 0,

(1.9)

the Wronskian of f and g, where f, g ∈ C ([0, ∞)). We also use the standard convention to abbreviate (with a slight abuse of notation) the operator of multiplication in L2 ((0, ∞); dx) by an element f ∈ L1loc ((0, ∞); dx) (and similarly in the higher-dimensional context with (0, ∞) replaced by an appropriate open set Ω ⊂ Rn later) by the same symbol f (rather than Mf , etc.). For additional notational conventions we refer to the paragraph at the end of this introduction.

Dirichlet-to-Neumann Maps and Applications

193

Then, the following results hold (with B1 (·) abbreviating the ideal of trace class operators): Theorem 1.1. Assume V ∈ L1 ((0, ∞); dx) and let z ∈ C\[0, ∞) with Im(z 1/2 ) > 0. Then, −1 −1 D N u H0,+ − zI+ − zI+ v, u H0,+ v ∈ B1 (L2 ((0, ∞); dx)) (1.10) and

−1 . D − zI+ v = 1 + z −1/2 det I+ + u H0,+

∞

dx sin(z 1/2 x)V (x)f+ (z, x)

0

= W (f+ (z, ·), φD (1.11) + (z, ·)) = f+ (z, 0), ∞ −1 . N det I+ + u H0,+ − zI+ dx cos(z 1/2 x)V (x)f+ (z, x) v = 1 + iz −1/2 0

=−

D f ′ (z, 0) W (f+ (z, ·), θ+ (z, ·)) = + 1/2 . 1/2 iz iz

(1.12)

Equation (1.11) is the modern formulation of the celebrated result due to Jost and Pais [34]. Performing calculations similar to Section 4 in [24] for the pair N N of operators H0,+ and H+ , one obtains the analogous result (1.12). We emphasize that (1.11) and (1.12) exhibit a spectacular reduction of a Fredholm determinant, that is, an infinite determinant (actually, a symmetrized perturbation determinant), associated with the trace class Birmann–Schwinger kernel of a one-dimensional Schr¨ odinger operator on the half-line (0, ∞), to a simple Wronski determinant of C-valued distributional solutions of (1.4). This fact goes back to Jost and Pais [34] (see also [24], [48], [49], [50, Sect. 12.1.2], [60], [61, Proposition 5.7], and the extensive literature cited in these references). The principal aim of this paper is to explore the extent to which this fact may generalize to higher dimensions. While a direct generalization of (1.11), (1.12) appears to be difficult, we will next derive a formula for the ratio of such determinants which indeed permits a natural extension to higher dimensions. For this purpose we introduce the boundary trace operators γD (Dirichlet trace) and γN (Neumann trace) which, in the current one-dimensional half-line situation, are just the functionals, / / C([0, ∞)) → C, C 1 ([0, ∞)) → C, (1.13) γN : γD : h → −h′ (0). g → g(0), D N N In addition, we denote by mD 0,+ , m+ , m0,+ , and m+ the Weyl–Titchmarsh mD D N N , H+ , H0,+ , and H+ , respectively, that is, functions corresponding to H0,+ 1/2 , mD 0,+ (z) = iz

mD + (z) =

′ f+ (z, 0)

f+ (z, 0)

mN 0,+ (z) = − ,

mN + (z) = −

1 = iz −1/2 , mD 0,+ (z)

(1.14)

f+ (z, 0) 1 =− ′ . f+ (z, 0) mD + (z)

(1.15)

194


In the case where V is real-valued, we briefly recall the spectral theoretic significance of mD + : It is a Herglotz function (i.e., it maps the open complex upper half-plane C+ analytically into itself) and the measure dρD + in its Herglotz repreD and hence encodes all sentation is then the spectral measure of the operator H+ D . Similarly, mD spectral information of H+ + also encodes all spectral information N N N since −1/mD of H+ + = m+ is also a Herglotz function and the measure dρ+ in N . its Herglotz representation represents the spectral measure of the operator H+ N In particular, dρD + (respectively, dρ+ ) uniquely determine V a.e. on (0, ∞) by the inverse spectral approach of Gelfand and Levitan [20] or Simon [59], [26] (see also Remling [56] and Section 6 in the survey [21]). Then we obtain the following result for the ratio of the perturbation determinants in (1.11) and (1.12): D Theorem 1.2. Assume V ∈ L1 ((0, ∞); dx) and let z ∈ C\σ(H+ ) with Im(z 1/2 ) > 0. Then, −1 . N det I+ + u H0,+ − zI+ v −1 . D det I+ + u H0,+ − zI+ v D − zI )−1 V γ (H N − zI )−1 ∗ (1.16) = 1 − γN (H+ + D + 0,+

=

W (f+ (z), φN + (z)) 1/2 iz W (f+ (z), φD + (z))

=

′ f+ (z, 0) 1/2 iz f+ (z, 0)

=

mN mD 0,+ (z) + (z) . = D m0,+ (z) mN + (z)

(1.17)

The proper multi-dimensional generalization to Schr¨ odinger operators in L2 (Ω; dn x) corresponding to an open set Ω ⊂ Rn with compact, nonempty boundary ∂Ω then involves the operator-valued generalization of the Weyl–Titchmarsh D function mD + (z), the Dirichlet-to-Neumann map denoted by MΩ (z). In particular, we will derive the following multi-dimensional extension of (1.16) and (1.17) in Section 4: = D N Theorem 1.3. Assume Hypothesis 2.6 and let z ∈ C σ HΩD ∪σ H0,Ω ∪σ H0,Ω . Then, N −1 . − zIΩ v det 2 IΩ + u H0,Ω −1 . D det 2 IΩ + u H0,Ω − zIΩ v D −1 . N − zI )−1 ∗ etr(T2 (z)) (1.18) V γD (H0,Ω = det 2 I∂Ω − γN HΩ − zIΩ Ω D D −1 tr(T2 (z)) = det 2 MΩ (z)M0,Ω (z) e . (1.19)

Here, det2 (·) denotes the modified Fredholm determinant in connection with Hilbert–Schmidt perturbations of the identity, T2 (z) is given by D −1 D −1 ∗ −1 N T2 (z) = γN H0,Ω − zIΩ V HΩ − zIΩ , (1.20) V γD H0,Ω − zIΩ


195

and IΩ and I∂Ω represent the identity operators in L2 (Ω; dn x) and L2 (∂Ω; dn−1 σ), respectively (with dn−1 σ the surface measure on ∂Ω). For pertinent comments on the principal reduction of (a ratio of) modified Fredholm determinants associated with operators in L2 (Ω; dn x) on the left-hand side of (1.18) to a modified Fredholm determinant associated with operators in L2 (∂Ω; dn−1 σ) on the right-hand side of (1.18) and especially, in (1.19), we refer to Section 4. Finally, we briefly list some of the notational conventions used throughout this paper. Let T be a linear operator mapping (a subspace of) a Banach space into another, with dom(T ) and ran(T ) denoting the domain and range of T . The closure of a closable operator S is denoted by S. The kernel (null space) of T is denoted by ker(T ). The spectrum and resolvent set of a closed linear operator in a separable complex Hilbert space H (with scalar product denoted by (·, ·)H , assumed to be linear in the second factor) will be denoted by σ(·) and ρ(·). The Banach spaces of bounded and compact linear operators in H are denoted by B(H) and B∞ (H), respectively. Similarly, the Schatten–von Neumann (trace) ideals will subsequently be denoted by Bp (H), p ∈ N. Analogous notation B(H1 , H2 ), B∞ (H1 , H2 ), etc., will be used for bounded, compact, etc., operators between two Hilbert spaces H1 and H2 . In addition, tr(T ) denotes the trace of a trace class operator T ∈ B1 (H) and detp (IH + S) represents the (modified) Fredholm determinant associated with an operator S ∈ Bp (H), p ∈ N (for p = 1 we omit the subscript 1). Moreover, X1 ֒→ X2 denotes the continuous embedding of the Banach space X1 into the Banach space X2 . For general references on the theory of modified Fredholm determinants we refer, for instance, to [16, Sect. XI.9], [27, Chs. IX, XI], [28, Sect. IV.2], [58], and [61, Ch. 9].

2. Schr¨ odinger operators with Dirichlet and Neumann boundary conditions D In this section we primarily focus on various properties of Dirichlet, H0,Ω , and N Neumann, H0,Ω , Laplacians in L2 (Ω; dn x) associated with open sets Ω ⊂ Rn , n = 2, 3, introduced in Hypothesis 2.1 below. In particular, we study mapping −q D,N − zIΩ , q ∈ [0, 1], (IΩ the identity operator in L2 (Ω; dn x)) properties of H0,Ω −q D,N − zIΩ , f ∈ Lp (Ω; dn x), for and trace ideal properties of the maps f H0,Ω D −r N −s , and γD H0,Ω − zIΩ , for appropriate appropriate p ≥ 2, and γN H0,Ω − zIΩ r > 3/4, s > 1/4, with γN and γD being the Neumann and Dirichlet boundary trace operators defined in (2.2) and (2.3). At the end of this section we then introduce the Dirichlet and Neumann Schr¨ odinger operators HΩD and HΩN in L2 (Ω; dn x), that is, perturbations of the D N Dirichlet and Neumann Laplacians H0,Ω and H0,Ω by a potential V satisfying Hypothesis 2.6.

196

F. Gesztesy, M. Mitrea, and M. Zinchenko We start with introducing our assumptions on the set Ω:

Hypothesis 2.1. Let n = 2, 3 and assume that Ω ⊂ Rn is an open set with a compact, nonempty boundary ∂Ω. In addition, we assume that one of the following three conditions holds: (i) Ω is of class C 1,r for some 1/2 < r < 1; (ii) Ω is convex; (iii) Ω is a Lipschitz domain satisfying a uniform exterior ball condition. The class of domains described in Hypothesis 2.1 is a subclass of all Lipschitz domains with compact nonempty boundary. We also note that while ∂Ω is assumed to be compact, Ω may be unbounded (e.g., an exterior domain) in connection with conditions (i) or (iii). For more details in this context, in particular, for the precise definition of the uniform exterior ball condition, we refer to [23, App. A] and [25, App. A] (and the references cited therein, such as [30, Ch. 1], [31], [32], [35], [40, Ch. 3], [43], [44], [62, p. 189], [63], [68], and [70, Sect. I.4.2]). 0 (Dirichlet trace) by First, we introduce the boundary trace operator γD 0 γD u = u|∂Ω .

0 γD : C(Ω) → C(∂Ω),

(2.1)

Then there exists a bounded, linear operator γD (cf. [40, Theorem 3.38]), γD : H s (Ω) → H s−(1/2) (∂Ω) ֒→ L2 (∂Ω; dn−1 σ), γD : H

3/2

(Ω) → H

1−ε

2

n−1

(∂Ω) ֒→ L (∂Ω; d

σ),

1/2 < s < 3/2, ε ∈ (0, 1),

(2.2)

0 . That is, the two Dirichlet trace operawhose action is compatible with that of γD tors coincide on the intersection of their domains. We recall that dn−1 σ denotes the surface measure on ∂Ω and we refer to [23, App. A] for our notation in connection with Sobolev spaces (see also [40, Ch. 3], [68], and [70, Sect. I.4.2]). Next, we introduce the operator γN (Neumann trace) by

γN = ν · γD ∇ : H s+1 (Ω) → L2 (∂Ω; dn−1 σ),

1/2 < s < 3/2,

(2.3)

where ν denotes the outward pointing normal unit vector to ∂Ω. It follows from (2.2) that γN is also a bounded operator. Given Hypothesis 2.1, we introduce the self-adjoint and nonnegative Dirichlet D N and Neumann Laplacians H0,Ω and H0,Ω associated with the domain Ω as follows, D D = {u ∈ H 2 (Ω) | γD u = 0}, = −∆, dom H0,Ω (2.4) H0,Ω N 2 N H0,Ω = −∆, dom H0,Ω = {u ∈ H (Ω) | γN u = 0}. (2.5) D N and H0,Ω is provided in [23, App. A] (cf. also [55, A detailed discussion of H0,Ω Sects. X.III.14, X.III.15]). D N and H0,Ω introduced Lemma 2.2. Assume Hypothesis 2.1. Then the operators H0,Ω in (2.4) and (2.5) are nonnegative and self-adjoint in L2 (Ω; dn x) and the following boundedness properties hold for all q ∈ [0, 1] and z ∈ C\[0, ∞), −q N −q D , H0,Ω − zIΩ ∈ B L2 (Ω; dn x), H 2q (Ω) . (2.6) H0,Ω − zIΩ


197

The fractional powers in (2.6) (and in subsequent analogous cases) are defined via the functional calculus implied by the spectral theorem for self-adjoint operators. As explained in [23, Lemma A.2] (based on results in [33], [40, Thm. 4.4, App. B], [42], [45], [64, Chs. 1, 2], [65, Props. 4.5, 7.9], [66, Sect. 1.3, Thm. 1.18.10, Rem. 4.3.1.2], [69]), the key ingredients in proving Lemma 2.2 are the inclusions D N dom H0,Ω ⊂ H 2 (Ω), dom H0,Ω ⊂ H 2 (Ω) (2.7) and real interpolation methods. The next result is a slight extension of [23, Lemma 6.8] and provides an explicit discussion of the z-dependence of the constant c appearing in estimate (6.48) of [23]. For a proof we refer to [25]. Lemma 2.3 ([25]). Assume Hypothesis 2.1 and let 2 ≤ p, n/(2p) < q ≤ 1, f ∈ Lp (Ω; dn x), and z ∈ C\[0, ∞). Then, −q −q N D ∈ Bp L2 (Ω; dn x) , (2.8) − zIΩ , f H0,Ω − zIΩ f H0,Ω

and for some c > 0 (independent of z and f ) ; D ; ;f H0,Ω − zIΩ −q ;2 2 Bp (L (Ω;dn x))

|z|2q + 1 2 2 −q 2 ≤c 1+ D 2q (| · | − z) Lp (Rn ;dn x) f Lp(Ω;dn x) , dist z, σ H0,Ω ; N ; ;f H − zIΩ −q ;2 2 0,Ω Bp (L (Ω;dn x))

|z|2q + 1 2 2 −q 2 ≤c 1+ N 2q (| · | − z) Lp (Rn ;dn x) f Lp(Ω;dn x) . dist z, σ H0,Ω

(2.9)

(Here, in obvious notation, (| · |2 − z)−q denotes the function (|x|2 − z)−q , x ∈ Rn .) Next we recall certain boundedness properties of powers of the resolvents of Dirichlet and Neumann Laplacians multiplied by the Neumann and Dirichlet boundary trace operators, respectively: Lemma 2.4. Assume Hypothesis 2.1 and let ε > 0, z ∈ C\[0, ∞). Then, N − 1+ε D − 3+ε 4 4 γN H0,Ω , γD H0,Ω − zIΩ ∈ B L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) . − zIΩ (2.10) As in [23, Lemma 6.9], Lemma 2.4 follows from Lemma 2.2 and from (2.2) and (2.3). Corollary 2.5. Assume Hypothesis 2.1 and let f1 ∈ Lp1 (Ω; dn x), p1 ≥ 2, p1 > 2n/3, f2 ∈ Lp2 (Ω; dn x), p2 > 2n, and z ∈ C\[0, ∞). Then, denoting by f1 and f2 the operators of multiplication by functions f1 and f2 in L2 (Ω; dn x), respectively, one has N −1 γD H0,Ω − zIΩ f1 ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , (2.11)

198

F. Gesztesy, M. Mitrea, and M. Zinchenko D −1 γN H0,Ω − zIΩ f2 ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ)

(2.12)

and for some cj (z) > 0 (independent of fj ), j = 1, 2, ; −1 ; ; ; N − zI ≤ c1 (z) f1 Lp1 (Ω;dn x) , (2.13) f1 ; ; γD H0,Ω Ω Bp1 (L2 (Ω;dn x),L2 (∂Ω;dn−1 σ)) ; ; −1 ; ; D − zI ≤ c2 (z) f2 Lp2 (Ω;dn x) . (2.14) f2 ; ; γN H0,Ω Ω Bp2 (L2 (Ω;dn x),L2 (∂Ω;dn−1 σ))

As in [23, Corollary 6.10], Corollary 2.5 follows from Lemmas 2.3 and 2.4. Finally, we turn to our assumptions on the potential V and the corresponding definition of Dirichlet and Neumann Schr¨ odinger operators HΩD and HΩN in L2 (Ω; dn x): Hypothesis 2.6. Suppose that Ω satisfies Hypothesis 2.1 and assume that V ∈ Lp (Ω; dn x) for some p satisfying 4/3 < p ≤ 2, in the case n = 2, and 3/2 < p ≤ 2, in the case n = 3. Assuming Hypothesis 2.6, we next introduce the perturbed operators HΩD and HΩN in L2 (Ω; dn x) by alluding to abstract perturbation results due to Kato [36] (see also Konno and Kuroda [37]) as summarized in [23, Sect. 2]: Let V , u, and v denote the operators of multiplication by functions V , u = exp(i arg(V ))|V |1/2 , and v = |V |1/2 in L2 (Ω; dn x), respectively, such that V = uv.

(2.15)

Since u, v ∈ L2p (Ω; dn x), Lemma 2.3 yields −1/2 −1/2 D D v ∈ B2p L2 (Ω; dn x) , − zIΩ , H0,Ω − zIΩ u H0,Ω −1/2 N −1/2 N − zIΩ , H0,Ω − zIΩ v ∈ B2p L2 (Ω; dn x) , u H0,Ω

z ∈ C\[0, ∞), (2.16) z ∈ C\[0, ∞), (2.17)

and hence, in particular,

- . N N 1/2 = H 1 (Ω) ⊃ H 2 (Ω) ⊃ dom H0,Ω , (2.18) dom(u) = dom(v) ⊇ dom H0,Ω . - D 1/2 D dom(u) = dom(v) ⊇ H 1 (Ω) ⊃ H01 (Ω) = dom H0,Ω ⊃ dom H0,Ω . (2.19)

Moreover, (2.16) and (2.17) imply D −1 N −1 u H0,Ω − zIΩ − zIΩ v, u H0,Ω v ∈ Bp L2 (Ω; dn x) 2 ⊂ B2 L (Ω; dn x) ,

z ∈ C\[0, ∞). (2.20)

Utilizing (2.9) in Lemma 2.3 with −z > 0 sufficiently large, such that the B2p norms of the operators in (2.16) and (2.17) are less than 1, one concludes that the Hilbert–Schmidt norms of the operators in (2.20) are less than 1. Thus, applying [23, Thm. 2.3], one obtains the densely defined, closed operators HΩD and HΩN N D D ∩ dom(V ) and H0,Ω (which are extensions of H0,Ω + V defined on dom H0,Ω +V


199

N defined on dom H0,Ω ∩ dom(V ), respectively). In particular, the resolvent of HΩD (respectively, HΩN ) is explicitly given by −1 D −1 D −1 D = H0,Ω − zIΩ − H0,Ω − zIΩ v HΩ − zIΩ + , −1 −1 D −1 D × IΩ + u H0,Ω − zIΩ v u H0,Ω − zIΩ , z ∈ C\σ HΩD , (2.21) −1 N −1 N −1 N HΩ − zIΩ = H0,Ω − zIΩ − H0,Ω − zIΩ v + N −1 ,−1 N −1 × IΩ + u H0,Ω − zIΩ v u H0,Ω − zIΩ , N z ∈ C\σ HΩ . (2.22) + D,N −1 , D,N is guaranteed by v for z ∈ ρ HΩ Here invertibility of IΩ + u H0,Ω − zIΩ

arguments discussed, for instance, in [23, Sect. 2] and the literature cited therein. Although we will not explicitly use the following result in this paper, we feel it is of sufficient independent interest to be included at the end of this section: Lemma 2.7. Assume Ω satisfies Hypothesis 2.1 with n = 2, 3 replaced by n ∈ N, n ≥ 2, suppose that V ∈ Ln/2 (Ω; dn x), and let ⎧ (0, 1) if n = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [0, 3 ) if n = 3, 2 s∈ (2.23) ⎪ [0, 2) if n = 4, ⎪ ⎪ ⎪ ⎪ ⎩ [0, 2] if n ≥ 5. Then the operator

∗ V : H s (Ω) → H 2−s (Ω)

(2.24)

is well defined and bounded, in fact, it is compact.

Proof. The fact that the operator (2.24) is well defined along with the estimate V B(H s (Ω),(H 2−s (Ω))∗ ) ≤ C(n, Ω)V Ln/2 (Ω) ,

(2.25)

are direct consequences of standard embedding results (cf. [67, Sect. 3.3.1] for smooth domains and [68] for arbitrary (bounded or unbounded) Lipschitz domains). Once the boundedness of (2.24) has been established, the compactness follows from the fact that if Vj ∈ C0∞ (Ω) is a sequence of functions with the prop ∗ by (2.25) erty that Vj → V in Ln/2 (Ω), then Vj → V in B H s (Ω), H 2−s (Ω) j↑∞ j↑∞ 2−s ∗ s and each operator Vj : H (Ω) → H (Ω) is compact, by Rellich’s selection lemma (cf. [17] for smooth domains and [68] for arbitrary Lipschitz domains). Thus, the operator in (2.24) is compact as the operator norm limit of a sequence of compact operators.

200


3. Dirichlet and Neumann boundary value problems and Dirichlet-to-Neumann maps This section is devoted to Dirichlet and Neumann boundary value problems associated with the Helmholtz differential expression −∆ − z as well as the corresponding differential expression −∆ + V − z in the presence of a potential V , both in connection with the open set Ω. In addition, we provide a detailed discussion of D N , MΩD , and Neumann-to-Dirichlet maps, M0,Ω , MΩN , Dirichlet-to-Neumann, M0,Ω 2 n−1 in L (∂Ω; d σ). Denote by ∗ γ )N : u ∈ H 1 (Ω) ∆u ∈ H 1 (Ω) → H −1/2 (∂Ω) (3.1)

a weak Neumann trace operator defined by dn x ∇u(x) · ∇Φ(x) + ∆u, Φ ) γN u, φ =

(3.2)

Ω

1/2

1

for all φ ∈ H (∂Ω) and Φ ∈ H (Ω) such that γD Φ = φ. We note that this definition is independent of the particular extension Φ of φ, and that ) γN is an extension of the Neumann trace operator γN defined in (2.3). For more details we refer to [23, App. A]. We start with a basic result on the Helmholtz Dirichlet and Neumann boundary value problems: Theorem ([25]). Assume Hypothesis 2.1. Then for every f ∈ H 1 (∂Ω) and = 3.1 D the following Dirichlet boundary value problem, z ∈ C σ H0,Ω / 3/2 (Ω), uD (−∆ − z)uD 0 ∈H 0 = 0 on Ω, (3.3) γD u D = f on ∂Ω, 0 2 n−1 σ). Moreover, there exist γN uD has a unique solution uD 0 ∈ L (∂Ω; d 0 satisfying ) constants C D = C D (Ω, z) > 0 such that

D uD (3.4) 0 H 3/2 (Ω) ≤ C f H 1 (∂Ω) . N n−1 σ) and z ∈ C\σ H0,Ω the following Neumann Similarly, for every g ∈ L (∂Ω; d boundary value problem, / 3/2 (−∆ − z)uN uN (Ω), 0 = 0 on Ω, 0 ∈ H (3.5) N γ )N u0 = g on ∂Ω, 2

N 1 has a unique solution uN 0 satisfying γD u0 ∈ H (∂Ω). Moreover, there exist constants C N = C N (Ω, z) > 0 such that N uN 0 H 3/2 (Ω) ≤ C gL2(∂Ω;dn−1 σ) .

(3.6)

In addition, (3.3)–(3.6) imply that the following maps are bounded = D D −1 ∗ ∗ γN H0,Ω − zIΩ : H 1 (∂Ω) → H 3/2 (Ω), z ∈ C σ H0,Ω , (3.7) −1 ∗ ∗ N = N 2 n−1 3/2 γD H0,Ω − zIΩ : L (∂Ω; d σ) → H (Ω), z ∈ C σ H0,Ω . (3.8)

Dirichlet-to-Neumann Maps and Applications N Finally, the solutions uD 0 and u0 are given by the formulas D −1 ∗ D u0 (z) = − γN H0,Ω − zIΩ f, −1 ∗ N N g. u0 (z) = γD H0,Ω − zIΩ

201

(3.9) (3.10)

A detailed proof of Theorem 3.1 will appear in [25]. We temporarily strengthen our hypothesis on V and introduce the following assumption:

Hypothesis 3.2. Suppose the set Ω satisfies Hypothesis 2.1 and assume that V ∈ L2 (Ω; dn x) ∩ Lp (Ω; dn x) for some p > 2. By employing a perturbative approach, we now extend Theorem 3.1 in connection with the Helmholtz differential expression −∆ − z on Ω to the case of a Schr¨ odinger differential expression −∆ + V − z on Ω. 1 Theorem = 3.3. Assume Hypothesis 3.2. Then for every f ∈ H (∂Ω) and z ∈ C σ HΩD the following Dirichlet boundary value problem, / (−∆ + V − z)uD = 0 on Ω, uD ∈ H 3/2 (Ω), (3.11) γD uD = f on ∂Ω,

has a unique solution uD satisfying )N uD ∈ L2 (∂Ω; dn−1 σ). Similarly, for every = γ g ∈ L2 (∂Ω; dn−1 σ) and z ∈ C σ HΩN the following Neumann boundary value problem, / (−∆ + V − z)uN = 0 on Ω, uN ∈ H 3/2 (Ω), (3.12) γ )N uN = g on ∂Ω,

has a unique solution uN . Moreover, the solutions uD and uN are given by the formulas −1 ∗ ∗ f, (3.13) uD (z) = − γN HΩD − zIΩ −1 ∗ ∗ g. (3.14) uN (z) = γD HΩN − zIΩ D = D Proof. We temporarily assume that z ∈ C σ H0,Ω ∪ σ HΩ in the case of the = N Dirichlet problem and z ∈ C σ H0,Ω ∪ σ HΩN in the context of the Neumann problem. / σ(HΩN ), respectively. Uniqueness follows from the fact that z ∈ / σ(HΩD ) and z ∈ Next, we will show that the functions −1 D D uD (z) = uD V u0 (z), (3.15) 0 (z) − HΩ − zIΩ −1 N N uN (z) = uN (z) − H − zI V u (z), (3.16) Ω 0 Ω 0

N with uD 0 , u0 given by Theorem 3.1, satisfy (3.13) and (3.14), respectively. Indeed, N 3/2 2 n−1 (Ω) and γ )N uD σ). it follows from Theorem 3.1 that uD 0 , u0 ∈ H 0 ∈ L (∂Ω; d 3/2 q n Using the Sobolev embedding theorem H (Ω) ֒→ L (Ω; d x), q ≥ 2, and the

202


N 2 n fact that V ∈ Lp (Ω; dn x), p > 2, one concludes that V uD 0 , V u0 ∈ L (Ω; d x), and hence (3.15) and (3.16) are well defined. Since one also has V ∈ L2 (Ω; dn x), it D −1 −1 N follows from Lemma 2.3 that V H0,Ω − zIΩ and V H0,Ω − zIΩ are Hilbert– Schmidt, and hence −1 −1 = D D ∪ σ HΩD , − zIΩ ∈ B L2 (Ω; dn x) , z ∈ C σ H0,Ω I + V H0,Ω (3.17) −1 −1 N 2 = N N n ∈ B L (Ω; d x) , z ∈ C σ H0,Ω ∪ σ HΩ . I + V H0,Ω − zIΩ (3.18)

Thus, by (2.4) and (2.5), −1 D D −1 D −1 −1 D D I + V H0,Ω − zIΩ V u0 = H0,Ω − zIΩ V u0 ∈ H 2 (Ω), HΩ − zIΩ (3.19) −1 N N −1 N −1 −1 N N 2 I + V H0,Ω − zIΩ V u0 = H0,Ω − zIΩ V u0 ∈ H (Ω), HΩ − zIΩ (3.20) )N uD ∈ L2 (∂Ω; dn−1 σ). Moreover, and hence uD , uN ∈ H 3/2 (Ω) and γ −1 D D D (−∆ + V − z)uD = (−∆ − z)uD V u0 0 + V u0 − (−∆ + V − z) HΩ − zIΩ (−∆ + V − z)u

N

D = V uD 0 − IΩ V u0 = 0,

= (−∆ −

z)uN 0

+V

uN 0

N = V uN 0 − IΩ V u0 = 0,

(3.21) N −1 N V u0 − (−∆ + V − z) HΩ − zIΩ

and by (2.4), (2.5) and (3.17), (3.18) one also obtains, −1 D D γD uD = γD uD V u0 0 − γD HΩ − zIΩ D −1 D −1 −1 D = f − γD H0,Ω − zIΩ I + V H0,Ω − zIΩ V u0 = f, −1 N N N N γN u = ) ) V u0 γN u0 − ) γN HΩ − zIΩ N −1 N −1 −1 N I + V H0,Ω = g − γN H0,Ω − zIΩ − zIΩ V u0 = g.

(3.22)

(3.23)

(3.24)

Finally, (3.13) and (3.14) follow from (3.9), (3.10), (3.15), (3.16), and the resolvent identity, −1 −1 ∗ ∗ D − zIΩ V − γN H0,Ω f uD (z) = IΩ − HΩD − zIΩ D D −1 ∗ −1 ∗ ∗ V f = − γN H0,Ω − zIΩ IΩ − HΩ − zIΩ −1 ∗ ∗ = − γN HΩD − zIΩ f, (3.25) −1 −1 ∗ ∗ N uN (z) = IΩ − HΩN − zIΩ V γD H0,Ω − zIΩ g N −1 ∗ N −1 ∗ ∗ = γD H0,Ω − zIΩ IΩ − HΩ − zIΩ V g −1 ∗ ∗ N g. (3.26) = γD HΩ − zIΩ


203

Analytic continuation D with respect to z then permits one to remove the additional condition z ∈ / σ H0,Ω in the case of the Dirichlet problem, and the additional N condition z ∈ / σ H0,Ω in the context of the Neumann problem.

Assuming Hypothesis 3.2, we now introduce the Dirichlet-to-Neumann maps, D M0,Ω (z) and MΩD (z), associated with (−∆ − z) and (−∆ + V − z) on Ω, as follows, / = D H 1 (∂Ω) → L2 (∂Ω; dn−1 σ), D z ∈ C σ H0,Ω M0,Ω , (3.27) (z) : f → −) γN u D , 0 where uD 0 is the unique solution of

and

(−∆ − z)uD 0 = 0 on Ω, MΩD (z) :

/

3/2 uD (Ω), 0 ∈H

γD uD 0 = f on ∂Ω, = z ∈ C σ HΩD ,

H 1 (∂Ω) → L2 (∂Ω; dn−1 σ), f → −) γN u D ,

where uD is the unique solution of (−∆ + V − z)uD = 0 on Ω,

uD ∈ H 3/2 (Ω),

γD uD = f on ∂Ω.

(3.28)

(3.29)

(3.30)

In addition, still assuming Hypothesis 3.2, we introduce the Neumann-toN (z) and MΩN (z), associated with (−∆ − z) and (−∆ + V − z) Dirichlet maps, M0,Ω on Ω, as follows, / = N L2 (∂Ω; dn−1 σ) → H 1 (∂Ω), N M0,Ω (z) : z ∈ C σ H0,Ω , (3.31) g → γD uN , 0 where uN 0 is the unique solution of

and

(−∆ − z)uN 0 = 0 on Ω, MΩN (z) :

/

3/2 uN (Ω), 0 ∈ H

L2 (∂Ω; dn−1 σ) → H 1 (∂Ω), g → γD uN ,

γ )N uN 0 = g on ∂Ω,

where uN is the unique solution of (−∆ + V − z)uN = 0 on Ω,

uN ∈ H 3/2 (Ω),

= z ∈ C σ HΩN ,

γ )N uN = g on ∂Ω.

(3.32)

(3.33)

(3.34)

It follows from Theorems 3.1 and 3.3, that under the assumption of HypothD N esis 3.2, the operators M0,Ω (z), MΩD (z), M0,Ω (z), and MΩN (z) are well defined and satisfy the following equalities, N = D N D ∪ σ H0,Ω , (3.35) (z) = −M0,Ω (z)−1 , z ∈ C σ H0,Ω M0,Ω = MΩN (z) = −MΩD (z)−1 , z ∈ C σ HΩD ∪ σ HΩN , (3.36)

and

D −1 ∗ ∗ D M0,Ω (z) = γ )N γN H0,Ω − zIΩ ,

= D , z ∈ C σ H0,Ω

(3.37)

204

F. Gesztesy, M. Mitrea, and M. Zinchenko −1 ∗ ∗ = )N γN HΩD − zIΩ , z ∈ C σ HΩD , MΩD (z) = γ −1 ∗ ∗ N = N N , − zIΩ , z ∈ C σ H0,Ω (z) = γD γD H0,Ω M0,Ω = N −1 ∗ ∗ N N MΩ (z) = γD γD HΩ − zIΩ , z ∈ C σ HΩ .

(3.38) (3.39) (3.40)

The representations (3.37)–(3.40) provide a convenient point of departure for proving the operator-valued Herglotz property of MΩD and MΩN . We will return to this topic in a future paper. Next, we note that the above formulas (3.37)–(3.40) may be used as alternative definitions of the Dirichlet-to-Neumann and Neumann-to-Dirichlet maps. In particular, we will next use (3.38) and (3.40) to extend the above definition of the operators MΩD (z) and MΩN (z) to the more general situation governed by Hypothesis 2.6: Lemma 3.4 ([25]). Assume Hypothesis 2.6. Then the operators MΩD (z) and MΩN (z) defined by equalities (3.38) and (3.40) have the following boundedness properties, = MΩD (z) ∈ B H 1 (∂Ω), L2 (∂Ω; dn−1 σ) , z ∈ C σ HΩD , (3.41) = MΩN (z) ∈ B L2 (∂Ω; dn−1 σ), H 1 (∂Ω) , z ∈ C σ HΩN . (3.42)

A detailed proof of Lemma 3.4 will be provided in [25]. Weyl–Titchmarsh operators, in a spirit close to ours, have recently been discussed by Amrein and Pearson [2] in connection with the interior and exterior of a ball in R3 and real-valued potentials V ∈ L∞ (R3 ; d3 x). For additional literature on Weyl–Titchmarsh operators, relevant in the context of boundary value spaces (boundary triples, etc.), we refer, for instance, to [1], [3], [4], [5], [6], [7], [14], [15], [22], [29, Ch. 3], [38], [39], [41], [52], [53]. Next, we prove the following auxiliary result, which will play a crucial role in Theorem 4.2, the principal result of this paper. Lemma 3.5. Assume Hypothesis 2.6. Then the following identities hold, −1 D −1 ∗ ∗ D )N HΩD − zIΩ V γN H0,Ω − zIΩ , (z) − MΩD (z) = γ M0,Ω = D z ∈ C σ H0,Ω ∪ σ HΩD , −1 N −1 ∗ ∗ D )N HΩD − zIΩ (z)−1 = I∂Ω − γ V γD H0,Ω − zIΩ , MΩD (z)M0,Ω = D D N z ∈ C σ H0,Ω ∪ σ HΩ ∪ σ H0,Ω . D = D Proof. Let z ∈ C σ H0,Ω ∪ σ HΩ . Then (3.43) follows from (3.37), and the resolvent identity D −1 D −1 ∗ ∗ D (z) − MΩD (z) = ) γN γN H0,Ω − zIΩ − HΩ − zIΩ M0,Ω −1 D −1 ∗ ∗ =γ )N γN HΩD − zIΩ V H0,Ω − zIΩ −1 D −1 ∗ ∗ )N HΩD − zIΩ V γN H0,Ω − zIΩ . =γ

(3.43)

(3.44) (3.38),

(3.45)


205

= D N Next, if z ∈ C σ H0,Ω ∪ σ HΩD ∪ σ H0,Ω , then it follows from (3.35), (3.39), and (3.43) that D D D MΩD (z)M0,Ω (z)−1 = I∂Ω + MΩD (z) − M0,Ω (z) M0,Ω (z)−1 D N = I∂Ω + M0,Ω (z) − MΩD (z) M0,Ω (z) D −1 D −1 ∗ ∗ )N HΩ − zIΩ V γN H0,Ω − zIΩ (3.46) = I∂Ω + γ N −1 ∗ ∗ × γD γD H0,Ω − zIΩ . Let g ∈ L2 (∂Ω; dn−1 σ). Then by Theorem 3.1, −1 ∗ ∗ N − zIΩ g u = γD H0,Ω

(3.47)

is the unique solution of

(−∆ − z)u = 0 on Ω,

u ∈ H 3/2 (Ω),

γN u = g on ∂Ω. )

(3.48)

Setting f = γD u ∈ H 1 (∂Ω) and utilizing Theorem 3.1 once again, one obtains −1 ∗ D u = − γN H0,Ω f − zIΩ −1 ∗ ∗ N −1 ∗ ∗ D γD γD H0,Ω − zIΩ g. (3.49) = − γN H0,Ω − zIΩ

Thus, it follows from (3.47) and (3.49) that D −1 ∗ ∗ N −1 ∗ ∗ γN H0,Ω − zIΩ γD γD H0,Ω − zIΩ −1 ∗ ∗ N = − γD H0,Ω − zIΩ .

(3.50)

Finally, insertion of (3.50) into (3.46) yields (3.44).

It follows from (4.25)–(4.30) that γ )N can be replaced by γN on the right-hand side of (3.43) and (3.44).

4. A multi-dimensional variant of a formula due to Jost and Pais In this section we prove our multi-dimensional variants of the Jost and Pais formula as discussed in the introduction. We start with an elementary comment on determinants which, however, lies at the heart of the matter of our multi-dimensional variant of the one-dimensional Jost and Pais result. Suppose A ∈ B(H1 , H2 ), B ∈ B(H2 , H1 ) with AB ∈ B1 (H2 ) and BA ∈ B1 (H1 ). Then, det(IH2 − AB) = det(IH1 − BA).

(4.1)

In particular, H1 and H2 may have different dimensions. Especially, one of them may be infinite and the other finite, in which case one of the two determinants in (4.1) reduces to a finite determinant. This case indeed occurs in the original one-dimensional case studied by Jost and Pais [34] as described in detail in [24] and the references therein. In the proof of the next theorem, the role of H1 and H2 will be played by L2 (Ω; dn x) and L2 (∂Ω; dn−1 σ), respectively.

206


We start with an extension of a result in [23]: = D N Theorem 4.1. Assume Hypothesis 2.6 and let z ∈ C σ HΩD ∪σ H0,Ω ∪σ H0,Ω . Then, D −1 ∗ −1 D −1 N γN H0,Ω − zIΩ ∈ B1 L2 (∂Ω; dn−1 σ) , V HΩ − zIΩ V γD H0,Ω − zIΩ (4.2) D −1 N −1 ∗ 2 n−1 γN HΩ − zIΩ V γD H0,Ω − zIΩ ∈ B2 L (∂Ω; d σ) , (4.3) and

−1 . N − zIΩ v det 2 IΩ + u H0,Ω −1 . D det 2 IΩ + u H0,Ω − zIΩ v D −1 ∗ . −1 N (4.4) V γD H0,Ω − zIΩ = det 2 I∂Ω − γN HΩ − zIΩ .. - - −1 ∗ −1 −1 D − zI N − zI . V HΩD − zIΩ V γD H0,Ω × exp tr γN H0,Ω Ω Ω

Proof. From the = outset that the left-hand side of (4.4) is well defined by we note D N ∪ σ H0,Ω and (2.20). Let z ∈ C σ HΩD ∪ σ H0,Ω u(x) = exp(i arg(V (x)))|V (x)|1/2 ,

p/p1

where

u )(x) = exp(i arg(V (x)))|V (x)| p1 =

/ 3p/2, n = 2, 4p/3, n = 3,

v(x) = |V (x)|1/2 ,

v)(x) = |V (x)|p/p2 ,

,

p2 =

/

3p, n = 2, 4p, n = 3

(4.5) (4.6)

(4.7)

with p as introduced in Hypothesis 2.6. Then it follows that p11 + p12 = p1 , in both cases n = 2, 3, and hence V = uv = u )v). Next, we introduce D −1 N −1 − zIΩ − zIΩ v, KN (z) = −u H0,Ω v (4.8) KD (z) = −u H0,Ω and note that

[IΩ − KD (z)]−1 ∈ B L2 (Ω; dn x) ,

Thus, utilizing the following facts,

z∈C

= D D . σ HΩ ∪ σ H0,Ω

[IΩ − KD (z)]−1 = IΩ + KD (z)[IΩ − KD (z)]−1 and

(4.9)

(4.10)

1 = det 2 [IΩ − KD (z)][IΩ − KD (z)]−1 (4.11) = det 2 IΩ − KD (z) det 2 [IΩ − KD (z)]−1 exp tr KD (z)2 [IΩ − KD (z)]−1 ,


207

one obtains det 2 [IΩ − KN (z)][IΩ − KD (z)]−1 = det 2 IΩ − KN (z) det 2 [IΩ − KD (z)]−1 × exp tr KN (z)KD (z)[IΩ − KD (z)]−1 (4.12) det 2 IΩ − KN (z) −1 exp tr (KN (z) − KD (z))KD (z)[IΩ − KD (z)] = . det 2 IΩ − KD (z)

At this point, the left-hand side of (4.4) can be rewritten as

N −1 . det 2 IΩ + u H0,Ω − zIΩ v det 2 IΩ − KN (z) = . −1 D det 2 IΩ − KD (z) det 2 IΩ + u H0,Ω − zIΩ v = det 2 [IΩ − KN (z)][IΩ − KD (z)]−1 × exp tr (KD (z) − KN (z))KD (z)[IΩ − KD (z)]−1 = det 2 IΩ + (KD (z) − KN (z))[IΩ − KD (z)]−1 × exp tr (KD (z) − KN (z))KD (z)[IΩ − KD (z)]−1 .

(4.13)

Next, temporarily suppose that V ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x). Using [23, Lemma A.3] (an extension of a result of Nakamura [47, Lemma 6]) and [23, Remark A.5], one finds −1 N −1 D KD (z) − KN (z) = −u H0,Ω − zIΩ − H0,Ω − zIΩ v −1 ∗ D −1 N γN H0,Ω − zIΩ v = −u γD H0,Ω − zIΩ , + ∗ −1 −1 N − zI D − zI u γN H0,Ω v. = − γD H0,Ω Ω Ω

(4.14)

Thus, inserting (4.14) into (4.13) yields,

−1 . N − zIΩ v det 2 IΩ + u H0,Ω D −1 . det 2 IΩ + u H0,Ω − zIΩ v + −1 ,∗ D −1 N − zI u γN H0,Ω − zIΩ v = det 2 IΩ − γD H0,Ω Ω . + , −1 D −1 × IΩ + u H0,Ω − zIΩ v - -+ −1 ,∗ D −1 N − zI u γN H0,Ω − zIΩ v × exp tr γD H0,Ω Ω −1 + −1 ,−1 .. D D . × u H0,Ω − zIΩ v IΩ + u H0,Ω − zIΩ v

(4.15)

208


Then, utilizing Corollary 2.5 with p1 and p2 as in (4.7), one finds, N −1 γD H0,Ω u ∈ Bp1 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) , − zIΩ D −1 γN H0,Ω − zIΩ v ∈ Bp2 L2 (Ω; dn x), L2 (∂Ω; dn−1 σ) ,

and hence, + −1 ,∗ D −1 N − zI γD H0,Ω u γN H0,Ω − zIΩ v ∈ Bp L2 (Ω; dn x) Ω ⊂ B2 L2 (Ω; dn x) , D −1 ,∗ −1 + N γN H0,Ω u ∈ Bp L2 (∂Ω; dn−1 σ) − zIΩ v γD H0,Ω − zIΩ ⊂ B2 L2 (∂Ω; dn−1 σ) . Moreover, using the fact that + −1 ,−1 D IΩ + u H0,Ω − zIΩ v ∈ B L2 (Ω; dn x) ,

z∈C

(4.16) (4.17)

(4.18)

(4.19)

D = D , σ HΩ ∪ σ H0,Ω (4.20)

one now applies the idea expressed in formula (4.1) and rearranges the terms in (4.15) as follows: N −1 . det 2 IΩ + u H0,Ω − zIΩ v −1 . D det 2 IΩ + u H0,Ω − zIΩ v D −1 ,−1 −1 + D − zIΩ v v IΩ + u H0,Ω = det 2 I∂Ω − γN H0,Ω − zIΩ ,∗ . + −1 N − zI u × γD H0,Ω Ω - - −1 −1 D − zI D − zI v u H0,Ω v × exp tr γN H0,Ω Ω Ω + D −1 ,∗ .. −1 ,−1 + N γD H0,Ω − zIΩ u × IΩ + u H0,Ω − zIΩ v D −1 ,−1 D −1 + = det 2 I∂Ω − γN H0,Ω − zIΩ ) H0,Ω − zIΩ v) v) IΩ + u ,∗ . + −1 N − zI u ) × γD H0,Ω Ω - - −1 −1 D − zI D − zI ) H0,Ω v) u v) (4.21) × exp tr γN H0,Ω Ω Ω + −1 ,∗ .. −1 ,−1 + N D . ) H0,Ω − zIΩ γD H0,Ω − zIΩ u ) v) × IΩ + u In the last equality we employed the following simple identities,

V = uv = u )) v, + + −1 ,−1 −1 ,−1 D D v IΩ + u H0,Ω − zIΩ v u=) v I +u ) H0,Ω − zIΩ v) u ).

(4.22)

(4.23)


209

Utilizing (4.21) and the following resolvent identity, D D D −1 −1 + −1 ,−1 HΩ − zIΩ ) H0,Ω v) = H0,Ω − zIΩ v) IΩ + u − zIΩ v ) , p

(4.24)

one arrives at (4.4), subject to the extra assumption V ∈ L (Ω; d x)∩L (Ω; dn x). Finally, assuming only V ∈ Lp (Ω; dn x) and utilizing [23, Thm. 3.2], Lemma 2.3, and Corollary 2.5 once again, one obtains + −1 ,−1 D IΩ + u ) H0,Ω − zIΩ v) ∈ B L2 (Ω; dn x) , (4.25) D −p/p1 2 u ) H0,Ω − zIΩ ∈ Bp1 L (Ω; dn x) , (4.26) D −p/p2 v) H0,Ω − zIΩ ∈ Bp2 L2 (Ω; dn x) , (4.27) N −1 2 γD H0,Ω − zIΩ u ) ∈ Bp1 L (Ω; dn x), L2 (∂Ω; dn−1 σ) , (4.28) D −1 2 γN H0,Ω − zIΩ v) ∈ Bp2 L (Ω; dn x), L2 (∂Ω; dn−1 σ) , (4.29) and hence

n

∞

−1 D u ) H0,Ω − zIΩ v ∈ Bp L2 (Ω; dn x) ⊂ B2 L2 (Ω; dn x) . )

(4.30)

Relations (4.24)–(4.30) prove (4.2) and (4.3), and hence, the left- and the righthand sides of (4.4) are well defined for V ∈ Lp (Ω; dn x). Thus, using (2.9), (2.13), (2.14), the continuity of det 2 (·) with respect to the Hilbert–Schmidt , the continuity of tr(·) with respect to the trace norm norm · 2 B2 L (Ω;dn x) , and an approximation of V ∈ Lp (Ω; dn x) by a sequence of po· B1 L2 (Ω;dn x)

tentials Vk ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x), k ∈ N, in the norm of Lp (Ω; dn x) as k ↑ ∞, then extends the result from V ∈ Lp (Ω; dn x) ∩ L∞ (Ω; dn x) to V ∈ Lp (Ω; dn x), n = 2, 3.

Given these preparations, we are now ready for the principal result of this paper, the multi-dimensional analog of Theorem 1.2: D N = ∪σ H0,Ω . Theorem 4.2. Assume Hypothesis 2.6 and let z ∈ C σ HΩD ∪σ H0,Ω Then, −1 D N − zI )−1 ∗ MΩD (z)M0,Ω (z)−1 − I∂Ω = −γN HΩD − zIΩ V γD (H0,Ω Ω ∈ B2 L2 (∂Ω; dn−1 σ) (4.31) and

N −1 . − zIΩ v det 2 IΩ + u H0,Ω −1 . D det 2 IΩ + u H0,Ω − zIΩ v D . −1 N − zI )−1 ∗ etr(T2 (z)) V γD (H0,Ω = det 2 I∂Ω − γN HΩ − zIΩ Ω D = det 2 MΩD (z)M0,Ω (z)−1 etr(T2 (z)) ,

(4.32) (4.33)

210


where

D −1 ∗ −1 D −1 N T2 (z) = γN H0,Ω − zIΩ V HΩ − zIΩ V γD H0,Ω − zIΩ ∈ B1 L2 (∂Ω; dn−1 σ) .

(4.34)

Proof. The result follows from combining Lemma 3.5 and Theorem 4.1. A few comments are in order at this point. The sudden appearance of the exponential term exp(tr(T2 (z))) in (4.32) and (4.33), when compared to the one-dimensional case, is due to the necessary use of the modified determinant detp (·) in Theorems 4.1 and 4.2. The multi-dimensional extension (4.32) of (1.16), under the stronger hypothesis V ∈ L2 (Ω; dn x), n = 2, 3, first appeared in [23]. However, the present results in Theorem 4.2 go decidedly beyond those in [23] in the sense that the class of domains Ω permitted by Hypothesis 2.1 is greatly enlarged as compared to [23] and the conditions on V satisfying Hypothesis 2.6 are nearly optimal by comparison with the Sobolev inequality (cf. Cheney [13], Reed and Simon [54, Sect. IX.4], Simon [57, Sect. I.1]). Moreover, the multi-dimensional extension (4.33) of (1.17) invoking Dirichlet-to-Neumann maps is a new result. The principal reduction in Theorem 4.2 reduces (a ratio of) modified Fredholm determinants associated with operators in L2 (Ω; dn x) on the left-hand side of (4.32) to modified Fredholm determinants associated with operators in L2 (∂Ω; dn−1 σ) on the right-hand side of (4.32) and especially, in (4.33). This is the analog of the reduction described in the one-dimensional context of Theorem 1.2, where Ω corresponds to the half-line (0, ∞) and its boundary ∂Ω thus corresponds to the one-point set {0}. In the context of elliptic operators on smooth k-dimensional manifolds, the idea of reducing a ratio of zeta-function regularized determinants to a calculation over the (k − 1)-dimensional boundary has been studied by Forman [18]. He also pointed out that if the manifold consists of an interval, the special case of a pair of boundary points then permits one to reduce the zeta-function regularized determinant to the determinant of a finite-dimensional matrix. The latter case is of course an analog of the one-dimensional Jost and Pais formula mentioned in the introduction (cf. Theorems 1.1 and 1.2). Since then, this topic has been further developed in various directions and we refer, for instance, to Burghelea, Friedlander, and Kappeler [8], [9], [10], [11], Carron [12], Friedlander [19], M¨ uller [46], Park and Wojciechowski [51], and the references therein. Remark 4.3. The following observation yields a simple application of formula (4.32). Since by the Birman–Schwinger = Dprinciple N(cf., e.g., the discussion in [23, Sect. 3]), for any z ∈ C σ HΩD ∪ σ H0,Ω ∪ σ H0,Ω , one has z ∈ σ HΩN if and N −1 . − zIΩ v = 0, it follows from (4.32) that only if det 2 IΩ + u H0,Ω D N = D , one has z ∈ σ HΩN (4.35) for all z ∈ C σ HΩ ∪ σ H0,Ω ∪ σ H0,Ω D −1 ∗ . −1 N = 0. V γD H0,Ω − zIΩ if and only if det 2 I∂Ω − γN HΩ − zIΩ


211

One can also prove the following analog of (4.32): −1 . D − zIΩ v det 2 IΩ + u H0,Ω N −1 . det 2 IΩ + u H0,Ω − zIΩ v D −1 N −1 ∗ . (4.36) V γD (HΩ − zIΩ )∗ = det 2 I∂Ω + γN H0,Ω − zIΩ - ∗ .. −1 −1 −1 D − zI N − zI . V HΩN − zIΩ V γD H0,Ω × exp − tr γN H0,Ω Ω Ω

Then, proceeding as before, one obtains N D = ∪ σ H0,Ω , one has z ∈ σ HΩD (4.37) for all z ∈ C σ HΩN ∪ σ H0,Ω D −1 N −1 ∗ ∗ . = 0. V γD HΩ − zIΩ if and only if det 2 I∂Ω + γN H0,Ω − zIΩ

Acknowledgments We are indebted to Yuri Latushkin and Konstantin A. Makarov for numerous discussions on this topic. We also thank the referee for a careful reading of our manuscript and his constructive comments. Fritz Gesztesy would like to thank all organizers of the international conference on Operator Theory and Mathematical Physics (OTAMP), and especially, Pavel Kurasov, for their kind invitation, the stimulating atmosphere during the meeting, and the hospitality extended to him during his stay in Lund in June of 2006. He also gratefully acknowledges a research leave for the academic year 2005/06 granted by the Research Council and the Office of Research of the University of Missouri–Columbia.

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Higher Derivatives of Spectral Functions Associated with One-dimensional Schr¨ odinger Operators D.J. Gilbert, B.J. Harris and S.M. Riehl Abstract. We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function, ρ(λ), on the positive real axis, in the context of one-dimensional Schr¨ odinger operators on the half-line with integrable potentials. In particular, we identify sufficient conditions on the potential for the existence and continuity of the nth derivative, ρ(n) (λ), and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. The potential relevance of our results to some topical issues in spectral theory is discussed. Mathematics Subject Classification (2000). 34B20, 34B24, 47B25, 47E05. Keywords. Sturm-Liouville problems, spectral functions, unbounded selfadjoint operators.

1. Introduction For the one-dimensional Schr¨ odinger operator H on the half-line, the associated spectral function, ρ(λ), and Titchmarsh-Weyl function, m(z), are well-known tools of classical spectral analysis. In particular ρ(λ) incorporates key information on the spectrum, such as its location and type, while on the upper half complex plane, m(z) is a Herglotz function which directly reflects the analyticity properties of the resolvent operator, (H − zI)−1 . These two functions are mutually related through the boundary properties of m(z), as the real axis is approached normally, and through the Stieltjes inversion formula. The m-function is also connected to solutions of the Schr¨ odinger equation in a number of ways. In the case of integrable potentials, the logarithmic derivative of the Jost solution of the Schr¨ odinger equation can be identified with a generalised Dirichlet m-function, and this analytic function can be continuously extended from the upper half-plane onto the positive real axis [13]. Moreover, given that the

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D.J. Gilbert, B.J. Harris and S.M. Riehl

one-dimensional Schr¨ odinger equation is of Sturm-Liouville type, the generalised m-function is a particular solution of a non-linear first order Riccati equation, which can sometimes be approximated for large λ using an iterative procedure (see, e.g., [6], [7]). These intricate relationships underlie the structure of this paper, which we now briefly outline. The selfadjoint operator H on L2 ([0, ∞)) with Dirichlet boundary condition at 0 is introduced in Section 2, and the main features of the spectrum, σ(H), are summarised for the case q ∈ L1 ([0, ∞)). We also collect together relevant properties of the spectral function and associated Titchmarsh-Weyl function and briefly discuss the relevance of the first and second derivatives of ρ(λ) to spectral concentration and resonances. In Section 3, sufficient conditions for the existence and continuity of the first n derivatives of the spectral density are identified in Theorem 1. Aspects of the relationship between the spectral density and a generalised Dirichlet m-function, m(x, z), are outlined in Section 4, where we also explore the possibility of using the Riccati equation to construct a series representation of m(x, z) which is valid for large z = λ ∈ R+ . In Section 5, we state our main result in Theorem 2. This identifies sufficient conditions on q to (M+1) ensure that ρ(M+1) (λ) ∼ ρ0 (λ) as λ → ∞, where ρ0 (λ) denotes the Dirichlet spectral function for H when q ≡ 0, and it follows that, under the conditions of the theorem, the M th derivative ρ(M) (λ) has no turning points for sufficiently large λ. We conclude with a brief discussion of the results and illustrate their potential application through an explicit example. The purpose of this paper is to provide an overview of some recent work by the authors on the differentiability of the spectral function. Full proofs of the main results will appear in a separate publication [8].

2. Mathematical background We consider the time-independent one-dimensional Schr¨ odinger operator H associated with the system Ly := −y ′′ + q(x)y = zy,

x ∈ [0, ∞), z ∈ C,

y(0) = 0, where the potential q is real valued and integrable on [0, ∞). In this case L is regular at x = 0 and in Weyl’s limit point case at infinity, so that for each z ∈ C\R there exists precisely one linearly independent solution of Lu = zu in L2 ([0, ∞)), and for each z ∈ R, there exists at most one such solution. The corresponding selfadjoint operator H acting on H = L2 ([0, ∞)) is defined by Hf = Lf, f ∈ D(H), where D(H) = {f ∈ H : Lf ∈ H; f, f ′ locally a.c.; f (0) = 0}.

Higher Derivatives of Spectral Functions

219

The essential spectrum σess (H) of H fills the semi-axis [0, ∞) and is purely absolutely continuous for λ > 0, while the negative spectrum, if any, consists of isolated eigenvalues, possibly accumulating at λ = 0. There may also be an eigenvalue at λ = 0, but only if xq(x) ∈ / L1 ([0, ∞)). In the context of this paper we are principally concerned with the absolutely continuous part of the spectrum on (0, ∞). The spectral function, ρ(λ), associated with H is a non-decreasing function on R, which we normalise in the usual way by setting ρ(0) = 0. There is a jump discontinuity in the spectral function at each eigenvalue, but otherwise ρ(λ) is constant on R− . For λ > 0, ρ(λ) is strictly increasing and continuously differentiable, with 1√ ρ′ (λ) ∼ λ as λ → ∞. π (see [2]). The spectrum, σ(H), may be defined as the complement in R of the set of points in a neighbourhood of which ρ(λ) is constant, and this is consistent with the more usual definition in terms of the resolvent operator. The spectrum of H may also be studied through properties of the TitchmarshWeyl m-function, m(z), which is closely related to the Green’s function for H and hence reflects the analyticity properties of the resolvent operator on C+ . The first derivative of ρ(λ) is known as the spectral density and is related to the boundary values of the m-function on R through the formula 1 (2.1) ρ′ (λ) = lim ℑm(λ + iǫ), ǫ→0 π which holds for all λ ∈ R for which the respective limits exist. Key features of the spectrum, such as its decomposition into absolutely continuous, singular continuous and pure point parts, may be extrapolated from knowledge of the spectral density on R, or equivalently, from the boundary behaviour of the m-function as z ∈ C+ approaches the real axis normally. The same spectral information may also be inferred from the asymptotic properties of solutions of Lu = λu, λ ∈ R, through the theory of subordinacy [9]. The investigation of more subtle features of the spectrum, such as spectral concentration and the associated issue of resonances, can benefit from information on the behaviour of the second derivative of the spectral function, as we now explain. We adopt the following definition of points of spectral concentration (cf. [1]). Definition 1 The point λc ∈ R is said to be a point of spectral concentration of H if (i) ρ′ (λ) exists finitely and is continuous in a neighbourhood of λc , and (ii) ρ′ (λ) has a local maximum at λc . In the present context, where q ∈ L1 ([0, ∞)), the definition effectively restricts attention to points of spectral concentration which occur in the interior of the absolutely continuous spectrum, σac (H), so that points of spectral concentration in the above sense can only arise on (0, ∞). As noted by Eastham [3], a consequence of the definition is that if ρ′′ (λ) exists and has one sign for λ > M > 0, then

220


ρ′ (λ) exists and is absolutely continuous for such λ, but has no local maximum in (M, ∞). Thus knowledge of the large λ behaviour of the second derivative of ρ(λ) provides a mechanism for estimating upper bounds for points of spectral concentration (see, e.g., [4], [6], [7]). Points of spectral concentration may be, but need not be, associated with resonances, which occur at poles of the meromorphic continuation of the Green’s function on the so-called unphysical sheet. There is an extensive literature on these phenomena, and the relationships between them (see, e.g., [10], [12], [15]). We will return to the issue of the contribution of higher derivatives of the spectral function to our understanding of the spectrum in Sections 5 and 6.

3. Sufficient conditions for higher derivatives In this section we identify sufficient conditions on the potential for the existence and continuity of the nth derivative of the spectral density on R+ . The proof is based on classical methods of Titchmarsh and uses the following generalisation of Gronwall’s inequality. Lemma 1. Let f (x) ≥ 0, g(x) ≥ 0, with f (x) continuous and (1 + x)n g(x) ∈ L1 ([0, ∞)). If C > 0 is constant and x f (t)g(t)dt f (x) ≤ C(1 + x)n + 0

for x ≥ 0, then

f (x) ≤ C(1 + x)n e

for x ≥ 0.

x (1+t)n g(t)dt 0

Let ρ(n+1) (λ) denote the nth derivative of the spectral density, ρ′ (λ). Theorem 1 demonstrates a neat correlation between the finiteness of the nth moment of q and the existence of the first n derivatives of the spectral density. Theorem 1. If q(x) is real valued and satisfies (x + 1)n q(x) ∈ L1 ([0, ∞)), then ρ(k+1) (λ) exists and is continuous for λ > 0; k = 0, 1, . . . , n. Idea of proof. The proof proceeds by induction and uses the Titchmarsh formula for the spectral density on R+ , viz. ρ′ (λ) = where a(λ)

:=

b(λ)

:=

1 √ , π λ(a2 (λ) + b2 (λ))

∞ √ 1 √ sin( λy)q(y)φ(y, λ)dy, λ 0 ∞ √ 1 1 √ +√ cos( λy)q(y)φ(y, λ)dy, λ λ 0


221

and φ(x, λ) is the solution of Lu = λu satisfying φ(0, λ) = 0, φ′ (0, λ) = 1. The result is first established for k = n, and the remaining cases follow immediately, since (x + 1)n q(x) ∈ L1 ([0, ∞)) implies (x + 1)k q(x) ∈ L1 ([0, ∞)) for k = 0, 1, 2, . . . , n − 1. Note that the Titchmarsh formula (see [14], Chapter V) is valid for all q ∈ L1 ([0, ∞)), and hence for all q in the scope of the theorem. The following corollary to Theorem 1 is immediate. Corollary 1. If q(x) = O(e−ax ) with a > 0, then ρ(n) (λ) is continuously differentiable for n = 0, 1, 2, . . . . We remark that the sufficient conditions of Theorem 1 are not in general necessary to ensure that the corresponding derivatives exist and are continuous for 1 1 sufficiently large λ. To see this, let n = 0 and q = sin((x + 1) 2 )(x + 1)− 2 . Then (x + 1)n q(x) = q(x) ∈ / L1 ([0, ∞)), but there exists Λ0 > 0 such that ρ(n+1) (λ) = ρ′ (λ) exists and is continuous on (Λ0 , ∞) (see [7], Example 2).

4. Series solutions of the Riccati equation In addition to its relationship to the spectral density, the Titchmarsh-Weyl mfunction is also linked to solutions of the differential equation. For z ∈ C+ , let χ(x, z) denote the so-called Jost solution of Lu = zu, which satisfies χ(x, z) ∼ exp (ikx) as x → ∞, where k 2 = z and the principal branch is chosen. Then the logarithmic derivative of χ(x, z) evaluated at x = 0 is equal to the value of the Dirichlet m-function at z, i.e., χ′ (x, z) = m(z), χ(x, z) x=0

where ′ denotes differentiation with respect to x. The asymptotic form of χ(x, z) implies that χ(x, z) ∈ L2 ([0, ∞); dx), and hence χ(x, z) cannot vanish at x = 0 for z ∈ C+ , since to do so would imply that H had a non-real eigenvalue at z, and thus contradict the selfadjointness of H. In a similar way, χ(x0 , z) = 0 for any x0 > 0, z ∈ C+ . We may therefore set m(x, z) :=

χ′ (x, z) χ(x, z)

(4.1)

+

for x ≥ 0, z ∈ C , and it is straightforward to check that for each x0 ≥ 0, m(x0 , z) is the Dirichlet m-function associated with the system: Lu = zu, x ≥ x0 , u(x0 , z) = 0. We will refer to m(x, z) as the generalised Dirichlet m-function, and note that in terms of our notation in Section 2, m(z) ≡ m(0, z). Moreover, since q ∈ L1 ([0, ∞)), the function m(x, z) may be continuously extended onto the positive real axis, z = λ ∈ R+ , for x ≥ 0. We then have (cf. (4.1)) that m(x, λ) is well defined, continuous and non-real for x ≥ 0, λ > 0, and satisfies m(x, λ) =

χ′ (x, λ) , χ(x, λ)

(4.2)

222


where χ(x, λ) ∈ L2 ([0, ∞); dx) is the pointwise limit as z ↓ λ of the Jost solution, and is itself a solution of Lu = λu [5], [13]. It follows from (4.2) and the well-known relationships between solutions of the Sturm-Liouville and Riccati equations that m(x, λ) satisfies the Riccati equation for x ≥ 0, λ > 0, so that

∂ 2 m(x, λ) = −λ + q(x) − (m(x, λ)) , (4.3) ∂x where m(x, λ) is the finite non-real limit as z ↓ λ of the generalised Dirichlet m-function, m(x, z). We now show that we can investigate the behaviour of the spectral density, ρ′0 (λ), using the Riccati equation (4.3). For x ≥ 0, λ > 0, we have m(x, λ) := ℜm(x, λ) + iℑm(x, λ), from which by (2.1), m(0, λ) := m(λ) = ℜm(λ) + iπρ′ (λ), so that for λ > 0

1 ℑm(0, λ). (4.4) π Thus in principle, ρ′ (λ) can be obtained for λ > 0 by finding the appropriate solution of the Riccati equation and evaluating at x = 0. If in addition it can be shown that m(x, λ) is differentiable n times with respect to λ for sufficiently large λ, we can also seek conditions under which ρ(n+1) (λ) exists and satisfies 1 , (4.5) ρ(n+1) (λ) = ℑm(n) (x, λ) π ρ′ (λ) =

x=0

where m(n) (x, λ) denotes the nth partial derivative of m(x, λ) with respect to λ. Equations (4.3) and (4.5) will form the basis for our investigation into the existence of upper bounds for turning points of higher derivatives of ρ(λ). The generalised Dirichlet m-function, m(x, λ), x ≥ 0, λ > 0, is a particular solution of (4.3) with known asymptotic behaviour as x, λ → ∞ for q ∈ L1 ([0, ∞)) (see, e.g., [13]). Since (4.3) is not in general solvable in terms of known functions, we proceed by postulating a series solution of the Riccati equation (4.3) of the form √ m(x, λ) = i λ + R(x, λ) + g(x, λ), (4.6) where g(x, λ) :=

∞

mn (x, λ),

n=0

with g(x, λ), R(x, λ) ∈ L1 ([0, ∞); dx), and g(x, λ), R(x, λ) → 0 as x, λ → ∞. These conditions on R and g ensure that the left-hand side of (4.6) has the required asymptotic behaviour and are sufficient to ensure uniqueness of the solution (see, e.g., [6], [13]). The aim is to construct a series representation of m(x, λ), which


223

is valid for all x ≥ 0 and sufficiently large λ, with a view to using this representation to obtain asymptotic estimates of ρ′ (λ) and its derivatives through the relationships (4.4) and (4.5). The mechanism through which the function R(x, λ) and the series g(x, λ) are obtained is as follows. We first substitute m(x, λ) from (4.6) into (4.3) to obtain 0 1 k−2 ∞ ∞ . . -√ m2k−1 + 2mk−1 m′k + 2 i λ + R mk = Q − m21 − mℓ , k=3

k=1

ℓ=1

√ where Q := q − R − R − 2i λ and ′ denotes differentiation with respect to x. This equation will be satisfied if solutions can be found to the following system of first order ordinary differential equations √ m′1 + 2(i λ + R)m1 = Q, √ m′2 + 2(i λ + R)m2 = −m21 , 1 0 k−2 √ ′ 2 k ≥ 3, mk + 2(i λ + R)mk = − mk−1 + 2mk−1 mℓ , ′

2

ℓ=1

which can be solved iteratively by quadrature to give ∞ √

t m1 (x, λ) = − e2i λ(t−x)+2 x R(s,λ)ds Q(t, λ)dt, x ∞ √

t m2 (x, λ) = e2i λ(t−x)+2 x R(s,λ)ds m21 (t, λ)dt,

(4.7) (4.8)

x

and for k ≥ 3, mk (x, λ) =

∞

x

√

e2i

λ(t−x)+2

t x

R(s,λ)ds

0

m2k−1 + 2mk−1

k−2 ℓ=1

mℓ

1

dt,

(4.9)

provided the integrals in (4.7), (4.8) and (4.9) converge. In addition to satisfying the asymptotic and convergence properties noted above, the function R(x, λ) is defined in such a way as to ensure that the derivatives R(M) (x, λ) and g (M) (x, λ) exist and are continuous for some fixed M ∈ N, where the superfix (M ) denotes the M th derivative with respect to λ. Note that the choice of R(x, λ), which determines both Q(x, λ) and g(x, λ), is not unique, and is also influenced by other considerations, for example a requirement that the series g(x, λ) be absolutely and uniformly convergent for all x ≥ 0 and sufficiently large λ. The process of determining a suitable choice of R(x, λ), which is itself dependent on q(x), has the effect of producing a set of conditions on q(x) which are sufficient to ensure that the generation of the series representation can be accomplished. These conditions contribute to the hypothesis of our main theorem in the following section.

224


5. Upper bounds for the turning points The main result in this section, Theorem 2, has direct implications for the existence and computability of upper bounds for turning points of higher derivatives of the spectral function, ρ(λ), for a significant class of integrable potentials. Note that the differentiability requirements in the hypothesis of the theorem are influenced by the choice of R(x, λ) in (5.5), while the inequalities in the hypothesis, together with (4.7) to (4.9) above, are used to establish the intermediate result in Lemma 2. Theorem 2. Let q(x) be continuously differentiable M −1 times, with the (M −1)th derivative q (M−1) (x) absolutely continuous on [0, ∞), for some M ≥ 2. Suppose that there exists a decreasing function φ(x) ∈ L1 ([0, ∞)), and a continuous nonnegative function ξ(λ) on (0, ∞) with ξ(λ) → 0 as λ → ∞, such that for x ≥ 0, (5.1) (x + 1)ℓ+1 q (ℓ) (x) ≤ φ(x) for ℓ = 0, 1, . . . , M − 1, ∞ (M) (t) dt ≤ (x + 1)−M+1 φ(x), (5.2) q x ∞ √ e2i λt q (M) (t)dt ≤ ξ(λ)(x + 1)−M+1 φ(x), (5.3) x

(ℓ)

where q (x) denotes the ℓth derivative of q. Then there exists ΛM ≥ 0 such that ρ(M+1) (λ) exists, is continuous on (ΛM , ∞), and satisfies

∞ .(M) (M+1) 1 -√ ≤ Cλ−M √1 + ξ(λ) ρ φ(t)dt (5.4) λ + ℑR(0, λ) (λ) − π λ 0 for all λ > ΛM , where

R(x, λ) :=

M

n

an (x)λ− 2 ,

(5.5)

n=1

with coefficients {an (x)} defined inductively by a1 (x)

:=

a2 (x)

:=

an+1 (x)

:=

i − q(x), 2 q ′ (x) , 4 0 i 2

a′n (x)

+

n−1 ℓ=1

1

an−ℓ (x)aℓ (x) ,

n = 2, 3, . . . , M − 1,

and the constant C depends on M and q, and is computable. Idea of proof. The proof of the theorem depends on the following intermediate result. Lemma 2. Let mj (x, λ), j = 1, 2, . . . , be as in (4.7)–(4.9), and suppose that the hypothesis of Theorem 2 is satisfied. Then for j = 1, 2, . . . , there exists ΛM ≥ 0


225

(M)

such that for λ > ΛM , the derivatives mj (x, λ) exist, are continuous and satisfy -√ . ∞ 1 (M) λ + ξ(λ) φ(t)dt mj (x, λ) ≤ K j λ−M 2 x

for all x ≥ 0, where the constant K depends on q and M , and is computable.

It may be inferred from the proof of the lemma that for sufficiently large λ, the series in (4.6) is uniformly and absolutely convergent in x and λ and that (4.6) may be differentiated term by term M times with respect to λ to give ∞ -√ .(M) m(M) (x, λ) = i λ + R(x, λ) + mn(M) (x, λ), (5.6) n=0

for λ > ΛM , from which it may be shown that (5.4) follows from (4.5) and Lemma 2. (k) Let ρ0 (λ) denote the kth derivative of the Dirichlet spectral function associated with Lu = zu for the case q(x) ≡ 0. The following corollary to Theorem 2 shows that if q(x) satisfies the conditions of the theorem, then ρ(k+1) (λ) approaches (k+1) ρ0 (λ) asymptotically as λ → ∞ for all k = 0, 1, . . . , M . Corollary 2. If the hypothesis of Theorem 2 is satisfied, then for k = 0, 1, . . . , M , there exists Λk > 0, such that ρ(k+1) (λ) exists for λ > Λk and 1 -√ .(k) (k+1) (λ) = λ (5.7) ρ(k+1) (λ) ∼ ρ0 π

as λ → ∞. In particular, ρ(k+1) (λ) eventually has one sign, namely the sign of the (k + 1)th derivative of ρ0 (λ).

Proof The result for k = 0 is already well known (see, e.g., [2]). The first part of the result for k = M > 0 follows easily from the asymptotic behaviour: 1(M) 0M

(M) k 1 R(0, λ)(M) = ak (0)λ− 2 , =O √ λ k=1 1 λ−M √ λ

=

λ−M ξ(λ)

=

(M) 1 , O √ λ

1 -√ .(M) , o √ λ λ

as λ → ∞. The second part of the result for k = M > 0 follows from the first, (M+1) taking into account from (5.4) that the sign of ρ0 (λ) is positive or negative for all λ according as M is odd or even respectively. To establish the result for k = 1, 2, . . . , M − 1, it is only necessary to observe that if the inequalities (5.1) to (5.3) hold for some M = M0 ≥ 2, then they also hold for M = 1, 2, . . . , M0 − 1.

226


Remark. Note that for given q and M satisfying the conditions of Theorem 2, the constant C is computable. It may then be inferred from the proof of Corollary 2 that for each k = 0, 1, 2, . . . , M − 1, there exists a computable constant, Bk , such (k+1) (λ) has one sign for λ > Bk . Such a constant is then an upper bound that ρ0 (k) for the turning points of ρ0 (λ).

6. Discussion and example We first discuss the relevance of our investigation to the long standing conjecture that a close correlation exists between the presence of resonances on the unphysical sheet and points of spectral concentration of ρ(λ) on R+ . This conjecture is particularly plausible if q is in L1 ([0, ∞)), since in this case the well-known Kodaira formula [11], viz. √ λ ρ′ (λ) = , λ > 0, π | χ(0, λ) |2 holds, while resonances occur at zeros of the Jost function χ(z) := χ(0, z) on the unphysical sheet, π ≤ arg k < 2π, where k 2 = z as in Section 4. However, evidence for a systematic correlation between the two phenomena is patchy and in the example below we see that even if a resonance lies arbitrarily close to the real axis, it need not give rise to a point of spectral concentration. The possibility remains that resonances which are insufficiently strong to induce points of spectral concentration may nonetheless leave some trace of their existence through points of inflection of ρ′ (λ), which are themselves turning points of ρ′′ (λ), or in the turning points of still higher derivatives. This is borne out by recent numerical and graphical experiments for the case of exponentially decaying potentials [16], and by the following example, in which the resonance and relevant turning point can be calculated explicitly. Example Let 2a2 , a > 0. (1 + ax)2 Then q ∈ L1 ([0, ∞)), and as noted in [5], the Jost solution of Lu = zu is given by

a , χ(x, z) = eikx 1 − ik(1 + ax) q=

with k 2 = z as before, so that by our remarks above, there is just one resonance, at k = −ia. Using (4.1) and (4.4), and recalling from Section 4 that we have set z = λ for z real and positive, yields for λ > 0, √ 3 λ(λ + 3a2 ) λ2 ′′ , so that ρ ρ′ (λ) = (λ) = > 0, λ > 0, π(λ + a2 ) 2π(λ + a2 )2 whence it follows from Definition 1 that there are no points of spectral concentration on (0, ∞). However, a further differentiation yields, in the notation of earlier


227

sections, ρ(3) (λ) =

3(λ + a2 )2 − 4λ(λ + 3a2 ) √ , 4π λ(λ + a2 )3

√ from which it may be inferred that ρ′ (λ) has a point of inflection at λ = (2 3 − 3)a2 ≃ (0.7a)2 , this being a global maximum of its derivative, ρ′′ (λ), on (0, ∞). The principal results reported in this paper, Theorems 1 and 2, are concerned with a wider and more systematic investigation into the existence and behaviour of higher derivatives of the spectral function ρ(λ), for q(x) ∈ L1 ([0, ∞)). In Theorem 1, we identify sufficient conditions on the potential to ensure that the nth derivative of the spectral density exists and is continuous for λ > 0. From the example above, we see that the conditions are not necessary, since in this case ρ(n) (λ) exists and is continuous on R+ for all n, even though the first moment of q is not finite. We note that Theorem 1 is sharp for n = 0, since there exist potentials, for example of von Neumann Wigner type, which just fail to be in L1 ([0, ∞)), and for which the spectral function has jump discontinuities on R+ . We suspect that the result is also sharp for n > 0, but this has not been established. The scope of Theorem 2 is both more limited and more wide ranging. Evidently, from the statement of the theorem, the conditions on q are quite restrictive and the conclusions only apply when λ is sufficiently large. On the other hand, if the hypothesis is satisfied, then for sufficiently large λ the existence and continuity of the derivatives is assured, the asymptotic behaviour of the derivatives as λ → ∞ is identified, and a mechanism for estimating upper bounds for the turning points of the derivatives is available. The two results are independent in the following sense. Suppose that q and n satisfy the conditions of Theorem 1 for n = N ; then the conditions of Theorem 2 may fail to be satisfied for the same q with M = N , and the converse situation also holds. To see this, consider the examples: N = 2, 1. q(x) = sin(1 + x)2 (1 + x)−4 , N = 2. 2. q(x) = sin(1 + x)−1 (1 + x)−3 , It is straightforward to check that Theorem 1 is applicable in Example 1, but Theorem 2 is not, while the reverse situation holds in Example 2. Acknowledgement The first author wishes to thank the European Commission for partial support under the INTAS programme, grant no. 05-1000008-7883.

References [1] B.M. Brown, M.S.P. Eastham and D.K.R. McCormack, Spectral concentration and perturbed discrete spectra, J. Comp. Appl. Math. 86 (1997), 415–425. [2] M.S.P. Eastham, The asymptotic nature of the spectral functions in Sturm-Liouville problems with continuous spectrum, J. Math. Anal. Appl. 213 (1997), 573–582.

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[3] M.S.P. Eastham, On the location of spectral concentration for Sturm-Liouville problems with rapidly decaying potential, Mathematika 45 (1998), 25–36. [4] M.S.P. Eastham, On the location of spectral concentration for perturbed discrete spectra, Mathematika 46 (1999), 145–154. [5] G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and their Applications Nova Science Publishers, New York, 2001. [6] D.J. Gilbert and B.J. Harris, Bounds for the points of spectral concentration of Sturm-Liouville problems, Mathematika 47, (2000), 327–337 (2002). [7] D.J. Gilbert, B.J. Harris and S.M. Riehl, The spectral function for Sturm-Liouville problems where the potential is of von Neumann-Wigner type or slowly decaying, J. Differential Equations 201, (2004), 139–159. [8] D.J. Gilbert, B.J. Harris and S.M. Riehl, to be published. [9] D.J. Gilbert and D.B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schr¨ odinger operators, J. Math. Anal. Appl. 128 (1987), 30–56. [10] P.D. Hislop and I.M. Segal, Introduction to Spectral Theory, with Applications to Schr¨ odinger Operators, Springer-Verlag, New York, 1996. [11] K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order, and Heisenberg’s theory of S-matrices, Amer. J. Math. 71 (1949), 921–945. [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, 1978. [13] A. Rybkin, Some new and old asymptotic representations of the Jost solution and the Weyl m-function for Schr¨ odinger operators on the line, Bull. London Math. Soc. 34 (2002), 61–72. [14] E.C. Titchmarsh, Eigenfunction Expansions I, 2nd edition, Oxford University Press, 1962. [15] E.C. Titchmarsh, Eigenfunction Expansions II, Oxford University Press, 1958. [16] N. White, private communication. D.J. Gilbert School of Mathematical Sciences Dublin Institute of Technology Kevin Street, Dublin 8, Ireland e-mail: [email protected] B.J. Harris Department of Mathematical Sciences Northern Illinois University Dekalb, IL 60115, USA e-mail: [email protected] S.M. Riehl Department of Mathematics University of Northern Iowa Cedar Falls, IA 50614-0506, USA e-mail: [email protected]


On the Boundary Value Problem for p-parabolic Equations Per-Anders Ivert Abstract. We derive a fundamental existence theorem for the equation u′t (x, t) − div(|∇x u|p−2 ∇x u) = 0 with continuous initial and boundary data in a space-time cylinder. Mathematics Subject Classification (2000). 35K15; 35K20; 35K65. Keywords. Degenerate parabolic equations, p-parabolic functions.

1. Introduction For the development of a potential theory for the operator u → P (u) = u′t (x, t) − div(|∇x u|p−2 ∇x u), it is of course important to have an existence theorem for solutions to the equation P (u) = 0 in some class of simple domains with continuous initial and boundary data. To the author’s knowledge, existence results published in the literature (see, e.g., [1]) depend on some regularity of these data, beyond continuity. In [6], a potential theory as mentioned above was discussed, and the existence result that we are going to derive here was anticipated but not proved there. The results of [1] do not seem applicable. A natural approach to the problem of securing existence of a solution under the mere assumption of continuity of boundary data, would be to approximate the data uniformly by smooth data and pass to the limit. To make this process work, one needs a preliminary existence theorem under such regularity assumptions on the data that correspond to regularity properties that can be deduced a priori for arbitrary solutions. This preliminary existence theorem is provided by [4]. The purpose of this work is to prove existence of a solution to the equation P (u) = 0 in a space-time cylinder, when continuous boundary values on the “parabolic boundary” of the cylinder are prescribed. Throughout the paper, we let Ω be an open subset of Rn with Lipschitz boundary and T a positive number.

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We put Z = Ω × (0, T ) ⊂ Rn+1 and denote the parabolic boundary of Z by ∂P Z, i.e., ∂P Z = (Ω × {0}) ∪ (∂Ω × [0, T ]).

Finally, the number p is strictly between 1 and ∞.

2. Preparatory estimates In this section, we adopt the following General assumptions: 1. u ∈ L∞ ((0, T ); L2 (Ω)), 2. f ∈ L1 (Z; Rn ), g ∈ L1 (Z; R), 3. For all ϕ ∈ C0∞ (Z) the following equality holds: ∂ϕ (x, t) dxdt u(x, t) − ∂t Z = g(x, t)ϕ(x, t) − f (x, t) · ∇x ϕ(x, t) dxdt.

(2.1)

Z

2.1. Weak continuity in L2 (Ω) We will show that under the general assumptions the mapping u : (0, T ) → L2 (Ω) has a canonical representative that is weakly continuous, i.e., continuous from the interval (0, T ) into the space L2 (Ω) with its weak topology. Let η be an increasing C ∞ -function on R with η(τ ) = 0 for τ ≤ 0 and η(τ ) = 1 for τ ≥ 1. Put, for h > 0 and τ ∈ R, T τ ηh (τ ) = η( ), uh (x, τ ) = u(x, t)ηh′ (t − τ ) dt, h 0 T T f (x, t)ηh′ (t − τ ) dt. g(x, t)ηh′ (t − τ ) dt, fh (x, τ ) = gh (x, τ ) = 0

0

Proposition 2.1. For every t ∈ (0, T ), the suite (uh (·, t))h>0 converges weakly in L2 (Ω) as h → 0+, and the limit u(·, t) is continuous as a mapping from (0, T ) into L2 (Ω) with the weak topology. Proof. Fix τ0 and τ with 0 < τ0 < τ < T . Choose ζ ∈ C0∞ (Ω). Choose, in (2.1), ϕ(x, t) = ηh (t − τ ) − ηk (t − τ0 ) ζ(x), where 0 < h < T − τ and 0 < k < T − τ0 : uh (x, τ ) − uk (x, τ0 ) ζ(x) dx − Ω = ηh (t − τ ) − ηk (t − τ0 ) g(x, t)ζ(x) − f (x, t) · ∇ζ(x) dxdt (2.2) Z

As (h, k) tends to (0, 0), the member on the right has the limit τ g(x, t)ζ(x) − f (x, t) · ∇ζ(x) dxdt − τ0

Ω

The Boundary Value Problem for p-parabolic Equations

231

Since the function t → u(·, t)L2 (Ω) is essentially bounded and L2 (Ω) is weakly complete, we conclude that the suites (uh (·, τ0 ))h>0 and (uh (·, τ ))h>0 converge weakly in L2 (Ω) as h → 0+. We thus get a canonical representative for u by letting u(·, t), for every t ∈ (0, T ), be the weak limit of uh (·, t) as h → 0+, and then t2 g(x, t)ζ(x) − f (x, t) · ∇ζ(x) dxdt u(x, t2 ) − u(x, t1 ) ζ(x) dx = t1

Ω

Ω

whenever 0 < t1 < t2 < T and ζ ∈ C0∞ (Ω). From this equation it is clear, because of the boundedness of u(·, t)L2 (Ω) , that the mapping t → u(·, t) is weakly continuous, i.e., continuous as a mapping from (0, T ) into L2 (Ω) with the weak topology. 2.2. Continuity in L2 (Ω) We now make the following Additional assumptions: 1. u ∈ Lp ((0, T ); W01,p (Ω)), ′ 2. f ∈ Lp (Z; Rn ), g ∈ L2 (Z; R).

We will show that under these Additional assumptions (and the previously made General assumptions) the mapping t → u(·, t) is continuous also with respect to the norm topology on L2 (Ω). Let 0 < h < T , fix t0 and t in (0, T − h) and let ζ ∈ C0∞ (Ω). We return to (2.2), put k = h, and differentiate both members with respect to τ : ∂uh (x, τ )ζ(x) dx = gh (x, τ )ζ(x) − fh (x, τ ) · ∇ζ(x) dx (2.3) Ω ∂τ Ω

This equality holds whenever 0 < τ < T − h, and by approximation we find that it holds for all ζ ∈ L2 (Ω) ∩ W01,p (Ω). We will prove Theorem 2.2. Under the General assumptions and the Additional assumptions the mapping t → u(·, t) is continuous from (0, T ) to L2 (Ω). Proof. Let 0 < t1 < t2 < T . According to Mazur’s lemma there are convex combinations Nν Uν = cν,j uhν,j , ν = 1, 2, . . . , j=1

of the functions uh such that Uν (·, t1 ) converges to u(·, t1 ) and Uν (·, t2 ) converges to u(·, t2 ) in the norm of L2 (Ω) as ν → ∞. More precisely, 0 < hν,j < T − t2 , cν,j ≥ 0,

lim

max hν,j = 0,

ν→∞ 1≤j≤Nν Nν

cν,j = 1,

j=1

lim u(·, ti ) − Uν (·, ti )L2 (Ω) = 0, i = 1, 2,

ν→∞

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where Uν (x, τ ) =

T

0

and Hν is defined by Hν =

Nν

u(x, t)Hν′ (t − τ ) dt

cν,j ηhν,j , ν = 1, 2, . . . .

j=1

We also put Gν (x, τ ) =

T 0

g(x, t)Hν′ (t − τ ) dt och Fν (x, τ ) =

0

T

f (x, t)Hν′ (t − τ ) dt.

Then, by linearity, (2.3) holds with uh , gh and fh replaced by Uν , Gν and Fν , respectively. We can, for every fixed t, choose ζ(x) = Uν (x, τ ) and then integrate between t1 and t2 . This yields Gν (x, τ )Uν (x, τ )−Fν (x, τ )·∇Uν (x, τ ) dx dτ. Uν (x, t2 )2 −Uν (x, t1 )2 dx = Ω

Ω

As ν → ∞ we get in the limit 1 u(x, t2 )2 − u(x, t1 )2 dx 2 Ω t2 g(x, t)u(x, t) − f (x, t) · ∇u(x, t) dx dt. = t1

(2.4)

Ω

This shows that the function τ → u(·, τ )L2 (Ω) is continuous, and since the function τ → u(·, τ ) is weakly continuous, we find that it is also strongly continuous. 2.3. The Hilbert transformation on R For a function f in the Schwarz class S(R) of rapidly decreasing functions we define its Fourier transform fˆ by ∞ fˆ(τ ) = f (t)e−2πitτ dt −∞

and its Hilbert transform f˜ by 1 f˜(t) = lim π εց0

|s−t|>ε

f (s) ds. t−s

In general, f˜ does not belong to S(R), but if f ∈ S(R), then f˜ ∈ C ∞ (R) and Dk f˜ ∈ Lp (R) for all k ∈ N and all p ∈ (1, ∞). The Fourier transforms fˆ of f and fˆ˜ of f˜ are related by ˆ f˜(τ ) = −i sgnτ fˆ(τ ).


233

Therefore, the Hilbert transformation has a unique isometric extension to L2 (R). It also has a unique continuous extension Lp (R) → Lp (R) for each p ∈ (1, ∞). If f ∈ L2 (R) and g ∈ L2 (R), Parseval’s formula gives ∞ ∞ ∞ g (t) dt = f (t)˜ fˆ(τ ) i sgnτ gˆ(τ ) dτ = − f˜(t)g(t) dt −∞

−∞

−∞

and, if f ∈ S(R) is real valued, ∞ ∞ f ′ (t)f˜(t) dt = 2πiτ fˆ(τ ) i sgnτ fˆ(τ ) dτ = − −∞

∞

−∞

−∞

2π|τ ||fˆ(τ )|2 dτ.

At the same time ∞ ∞ ∞ ∞ f (s) − f (t) 2 f (t + s) − f (t) 2 ds dt = dt ds s−t s −∞ −∞ −∞ −∞ ∞ ∞ 2πisτ − 1 2 ˆ 2 e = |f (τ )| dτ ds s −∞ −∞ ∞ 2πisτ ∞ − 1 2 e ˆ )|2 |f(τ = ds dτ s −∞ −∞ ∞ 4π 2 |τ ||fˆ(τ )|2 dτ. = −∞

Hence, ∞ f ′ (t)f˜(t) dt = − −∞

∞

1 2π|τ ||fˆ(τ )|2 dτ = 2π −∞

∞ −∞

f (s) − f (t) 2 ds dt, s−t −∞ ∞

and the latter of these two equalities holds also for all f ∈ L2 (R). 1

Definition 2.3. The space H 2 (R) consists of those elements f ∈ L2 (R) for which ∞ ∞ f (s) − f (t) 2 ds dt < ∞. s−t −∞ −∞ This space is normed by

f 21 = 2

∞

−∞

1 (1 + 4π 2 τ 2 ) 2 |fˆ(τ )|2 dτ.

1

2.4. An estimate in H 2 (R; L2 (Ω)) Let 0 < h < T and let θ be a C ∞ -function on R with supp θ ⊂ (0, T − h). Put ah (t, τ ) = θ(τ )ηh′ (t − τ ),

t ∈ R, τ ∈ R.

˜h (t, ·) the Hilbert transform of ah (t, ·). Let a ˆh (t, ·) be the Fourier transform and a As a test function ζ in (2.3) we choose, for each fixed τ , T u(x, s)˜ ah (s, τ ) ds. ζ(x) = θ(τ ) 0

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P.-A. Ivert

The left-hand member of (2.3) becomes T T - . − u(x, t)u(x, s) dx ηh′′ (t − τ )θ(τ )˜ ah (s, τ ) ds dt 0

0

=

T

0

−

0

Ω T

0 T

-

. ∂a h (t, τ )˜ ah (s, τ ) ds dt u(x, t)u(x, s) dx ∂τ Ω

θ′ (τ )uh (x, τ )u(x, s)˜ ah (s, τ ) dx ds.

Ω

From (2.3) we thus get T T - . ∂a h (t, τ )˜ ah (s, τ ) ds dt u(x, t)u(x, s) dx ∂t 0 0 Ω θ(τ ) gh (x, τ )u(x, s) − fh (x, τ ) · ∇x u(x, s) a ˜h (s, τ ) dx ds = Z

+

T

0

θ′ (τ )uh (x, τ )u(x, s)˜ ah (s, τ ) dx ds.

(2.5)

Ω

We will integrate both members of (2.5) with respect to t, but first we note that ∞ ∞ ∂ah (t, τ )˜ ah (s, τ ) dτ = ah (s, σ) dσ 2πiσˆ ah (t, σ) i sgnσˆ −∞ −∞ ∂τ ∞ ah (s, σ) dσ 2π|σ|ˆ ah (t, σ)ˆ =− −∞

=−

1 2π

1 2π 1 =− 2π

=−

Consequently, ∞ T −∞

0

0

Gh (x, t) =

∞

-

. ∂a h (t, τ )˜ ah (s, τ ) ds dt dτ u(x, t)u(x, s) dx ∂t Ω ∞ ∞ T u(x, t) ah (t, σ + τ ) − ah (t, σ) 2 dt dσ dτ dx τ Ω −∞ −∞ 0 ∞ ∞ θ(σ + τ )uh (x, σ + τ ) − θ(σ)uh (x, σ) 2 dσ dτ dx. τ Ω −∞ −∞

T

1 =− 2π 1 =− 2π

We now put

e2πiστ − 1 2 ah (s, σ) dτ dσ ˆh (t, σ)ˆ a τ −∞ −∞ ∞ ∞ 2πiστ e a ˆh (t, σ) − a ˆh (t, σ) e2πiστ a ˆh (s, σ) − a ˆh (s, σ) · dσ dτ τ τ −∞ −∞ ∞ ∞ ah (t, σ + τ ) − ah (t, σ) ah (s, σ + τ ) − ah (s, σ) · dσ dτ. τ τ −∞ −∞ ∞

∞

−∞

gh (x, τ )θ(τ )˜ ah (t, τ ) dτ, Fh (x, t) =

∞

−∞

fh (x, τ )θ(τ )˜ ah (t, τ ) dτ

The Boundary Value Problem for p-parabolic Equations and Bh (s, t) =

∞

−∞

θ′ (τ )˜ ah (t, τ )ηh (s − τ ) dτ.

When we integrate both members of (2.5) with respect to τ over R, we get ∞ ∞ 1 θ(t)uh (x, τ ) − θ(σ)uh (x, σ) 2 dσ dτ dx 2π Ω −∞ −∞ τ −σ = Fh (x, t) · ∇u(x, t) − Gh (x, t)u(x, t) dxdt Z

+

0

T

0

T

235

u(x, s)u(x, t)Bh (s, t) dx ds dt.

(2.6)

Ω

Now we let bh (x, ·) be the Hilbert transform of θ(·)gh (x, ·): ∞ Gh (x, t) = gh (x, τ )θ(τ )˜ ah (t, τ ) dτ −∞ ∞ bh (x, τ )ah (t, τ ) dτ = ηh′ ∗ θ(·)bh (x, ·) (t). =−

Since

ηh′ (τ )

≥ 0 and

∞

−∞

−∞

ηh′ (τ ) dt = 1 we get

Gh (x, ·)L2 (R) ≤ θ(·)bh (x, ·)L2 (R) ≤ max |θ(τ )|bh (x, ·)L2 (R) τ ∈R

= max |θ(τ )|θ(·)gh (x, ·)L2 (R) ≤ max |θ(τ )|2 g(x, ·)L2 (0,T ) , τ ∈R

τ ∈R

which leads to Gh L2 (Z) ≤ max |θ(τ )|2 gL2(Z) . τ ∈R

In the same way we get Fh Lp′ (R) ≤ C max |θ(τ )|2 f Lp′ (Z) , τ ∈R

where C is a constant, depending only on p. To estimate the last term in the righthand member of (2.6), we let ch (s, ·) be the Hilbert transform of θ′ (·)ηh (s−·). Then ∞ ah (t, τ )ch (s, τ ) dt = ηh′ ∗ θ(·)ch (s, ·) (τ ). Bh (s, t) = − −∞

Hence

Bh (s, ·)L2 (0,T ) ≤ θ(·)ch (s, ·)L2 (R) > ∞ |θ′ (τ )ηh (s − τ )|2 dτ ≤ max |θ(τ )| τ ∈R

−∞

≤ max |θ(τ )|θ′ L2 (R) . τ ∈R

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P.-A. Ivert

We arrive at

T

0

u(x, s)u(x, t)Bh (s, t) dx dt Ω > T - .2 ≤ Bh (s, ·)L2 (0,T ) u(x, t)u(x, s) dx dt Ω

0

≤ max |θ(τ )|θ′ L2 (R) τ ∈R

>

T

0

u(·, t)2L2 (Ω) u(·, s)2L2 (Ω) dt

= max |θ(τ )|θ′ L2 (R) uL2(Z) u(·, s)L2 (Ω) , τ ∈R

and finally T T √ u(x, s)u(x, t)Bh (s, t) dx dt ds ≤ T max |θ(τ )|θ′ L2 (R) u2L2(Z) . 0

0

τ ∈R

Ω

From (2.6) we now get ∞ ∞ 1 θ(τ )uh (x, τ ) − θ(σ)uh (x, σ) 2 dσ dτ dx 2π Ω −∞ −∞ τ −σ

≤ C max |θ(τ )|2 f Lp′ (Z) ∇uLp(Z) + max |θ(τ )|2 gL2(Z) uL2(Z) τ ∈R τ ∈R √ + T max |θ(τ )|θ′ L2 (R) u2L2(Z) . (2.7) τ ∈R

According to Fatou’s lemma, this inequality is preserved when we pass to the limit h → 0+. By suitably choosing θ as a function which equals 1 in the interval [δ, T − δ] for some δ ∈ (0, T2 ), we deduce the following Lemma 2.4. Let u be a solution to equation (2.1) and let the General and Additional assumptions be valid. Let 0 < δ < T2 . Then T −δ T −δ 1 u(x, t) − u(x, σ) 2 dσ dt dx 2π Ω δ t−σ δ 8 2T u2L2(Z) , ≤ C f Lp′ (Z) ∇uLp(Z) + gL2 (Z) uL2 (Z) + δ with a constant C that depends only on p.

3. The boundary value problem We will be concerned with the solvability of the equation ∂u = div(|∇u|p−2 ∇u), (x, t) ∈ Z ∂t under prescribed values on the parabolic boundary ∂P Z.

(3.1)


237

Definition 3.1. We say that u is a solution to equation (3.1) if the restriction of u to Ω′ × (δ, T − δ) belongs to L∞ ((δ, T − δ), L2 (Ω′ )) ∩ Lp ((δ, T − δ); W 1,p (Ω′ )) for every δ ∈ (0, T2 ) and every domain Ω′ such that Ω′ is a compact subset of Ω, and ∂ϕ (x, t) dxdt = u(x, t) |∇x u(x, t)|p−2 ∇x u(x, t) · ∇x ϕ(x, t) dxdt. (3.2) ∂t Z Z for all ϕ ∈ C0∞ (Z).

Our main result is Theorem 3.2. Let ψ be a continuous function on ∂P Z. Then there is a solution u to equation (3.1), continuous on Z, such that u(x, t) = ψ(x, t) on ∂P Z. 3.1. The function spaces B p (Q(+) ) Following Fontes [4] (with a slight modification of notation), we put u(x, s) − u(x, t) 2 B·p (Q) = {u ∈ Lp (R; W 1,p (Ω)); dx ds dt < ∞} s − t R R Ω

and

B0p (Q) = {u ∈ Lp (R; W01,p (Ω));

u(x, s) − u(x, t) 2 dx ds dt < ∞} s−t R R Ω

These two spaces are Banach spaces, normed by - .2/p u2B p = |u(x, t)|p + |∇x u(x, t)|p dxdt Q

+

1 2π

u(x, s) − u(x, t) 2 dx dsdt. s−t R R Ω

We put Q+ = Ω × [0, ∞). For functions u, defined in Q+ , we define the extension operators E0 and Ep by u(x, t) if t ≥ 0 E0 u(x, t) = Ep u(x, t) = u(x, |t|), 0 if t < 0,

and we also introduce the spaces

p B·,· (Q+ ) = {u ∈ Lp (R+ ; W 1,p (Ω)); Ep u ∈ B p (Q)},

p B0,· (Q+ ) = {u ∈ Lp (R+ ; W 1,p (Ω)); Ep u ∈ B0p (Q)},

p B∗,0 (Q+ ) = {u ∈ Lp (R+ ; W 1,p (Ω)); E0 u ∈ B∗p (Q)}.

In the last definition, ∗ stands optionally for · or 0. We also put ′ ′ ∂u p BIp (Q+ ) = {u ∈ B0,· ∈ Lp (R+ , W −1,p (Ω))} (Q+ ) ∩ Cb ([0, ∞), L2 (Ω)); ∂t p (Q+ ) + BIp (Q+ ). We will need the following result and X p (Q+ ) = B·,0 Lemma 3.3. Let g be a Lipschitz function on Z. Then g can be extended to Q+ in such a way that the extension belongs to X p (Q+ ).

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Proof. Extend g to Q+ in such a way that it satisfies a global Lipschitz condition, p vanishes for t large and belongs to B·,· (Q+ ) (which is obviously possible). Let η be a Lipschitz function on Ω, such that 0 ≤ η(x) ≤ 1, η(x) = 1 on ∂Ω and η(x) ≤ 1 − dist(x, ∂Ω) for x close to ∂Ω. Put 1

u1 (x, t) = η(x) t g(x, t), u2 (x, t) = g(x, t) − u1 (x, t).

p One can then verify that u1 ∈ B·,0 (Q+ ) and u2 ∈ BIp (Q+ ). The details are left to the reader.

3.2. Known results A standard result is the following: Lemma

3.4. If u is a bounded solution to (3.1) and K a compact subset of Z, then K |∇x u(x, t)|p dxdt can be estimated in terms of supZ u(x, t), the Lebesgue measure of K and the distance from K to the boundary of Z. The following result is a special case of Theorem 4.8 in [4]: Theorem 3.5. Given g ∈ X p (Q+ ), there exists a unique solution u ∈ X p (Q+ ) to p (Q+ ). equation (3.1), such that u − g ∈ B0,0 From the regularity theory for degenerate and singular parabolic equations (see [2]) it follows Theorem 3.6. If the function g in Theorem 3.5 is continuous, then the solution u is continuous on Q+ . A standard result is also the following comparison principle. Theorem 3.7. If u and v are solutions to equation (3.1) in Z, continuous on Z, and if u(x, t) ≤ v(x, t) on ∂P Z, then u(x, t) ≤ v(x, t) in Z. With the aid of the comparison principle, we can prove Lemma 3.8. Suppose that u, u1 , u2 , . . . are solutions to equation (3.1) in Z, continuous on Z, and assume that the sequence (uj |∂P Z)∞ j=1 converges uniformly on ∂P Z to u|∂P Z. Then the sequence (uj )∞ j=1 converges uniformly in Z to u. Proof. Let ε > 0. For j sufficiently large, u(x, t) − ε < uj (x, t) < u(x, t) + ε for all (x, t) ∈ ∂P Z. Since u−ε and u+ε are also solutions, it follows from the comparison principle that u(x, t) − ε < uj (x, t) < u(x, t) + ε for all (x, t) ∈ Z if j is sufficiently large.

4. Proof of the main theorem We can now prove Theorem 3.2. Let (ψj )∞ j=1 be a sequence of Lipschitz functions on Z, converging uniformly on ∂P Z to ψ. Each ψj can, by Lemma 3.3, be extended to a continuous element of X p (Q+ ). By Theorems 3.5 and 3.6, each ψj corresponds


239

to a continuous solution uj to (3.1), taking boundary values ψj . By the comparison principle, since max |ψj (x, t) − ψk (x, t)| = 0 lim j,k→∞ (x,t)∈∂P Z

we have lim

max |uj (x, t) − uk (x, t)| = 0

j,k→∞ (x,t)∈Z

and thus the sequence (uj )∞ j=1 converges uniformly on Z to a continuous function u. It remains to show that u is a solution. We let Ω′ be domain such that Ω′ is a compact subset of Ω and we let 0 < δ < T2 . It follows from Lemma 2.4 and Lemma 3.4 that the quantities T −δ T −δ T −δ 1 uj (x, τ ) − uj (x, σ) 2 |∇x uj (x, t)|p dt dx + dσ dτ dx 2π τ − σ ′ ′ Ω δ Ω δ δ are bounded uniformly in j, and thus the corresponding quantity for u is finite. We conclude that the restriction of u to Z ′ = Ω′ × (δ, T − δ) can be extended to an element of X p (Ω′ × (δ, ∞)) and thus there is a continuous solution v to equation (3.1) in Z ′ , taking the values u(x, t) on ∂P Z ′ . However, on ∂P Z ′ the sequence (uj )∞ j=1 converges uniformly to v(x, t) and thus, by Lemma 3.8, it converges uniformly to v in all of Z ′ . On the other hand, it converges to u(x, t), so u is indeed a solution to (3.1) in Z ′ . Since Ω′ is arbitrary and δ can be chosen arbitrarily small, we conclude that u is a solution to (3.1) in Z ′ , taking the boundary values ψ. This concludes the proof.

References [1] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z., 183 (1983), 311–341. [2] E. DiBenedetto, Degenerate Parabolic Equations. Springer Verlag, New York, 1993. [3] M. Fontes, A Monotone Operator Method for Elliptic-Parabolic Equations. Comm. PDE, 25, 3& 4, (2000), 681–702 [4] M. Fontes, Initial-boundary value problems for parabolic equations. Institut MittagLeffler, Report No. 22 (1999/2000). [5] M. Fontes, Parabolic Equations with Low regularity. Doctoral Thesis 1996:3, Lund University. [6] T. Kilpeläinen, P. Lindqvist, On the Dirichlet Boundary Value Problem for a Degenerate Parabolic Equation. Siam J. Math. Anal., Vol 27, No 3, (1996), 661–683. Per-Anders Ivert Centre for Mathematical Sciences Lund University S-221 00 LUND, Sweden e-mail: [email protected]


On the Rank One Dissipative Operator and the Parseval Formula Mitsuteru Kadowaki, Hideo Nakazawa and Kazuo Watanabe Abstract. The generalized Parseval formula concernig a non-selfadjoint operator is studied. In particular, the wave equation with the rank one dissipative term is considered. The existence of scattering and dissipative modes is obtained. Mathematics Subject Classification (2000). 35P10, 35P25. Keywords. Wave equation with dissipative term, dissipative mode, scattering mode, Parseval formula, wave operator, spectral singularity.

1. Introduction The purpose of the present article is to report on our recent results about the spectral problems of wave equations with a dissipative term. Section 2 and 3 were originally intended to be an announcement of results to be presented in [6] with full detail. Our main result is a generalized Parseval formula associated to dissipative wave equation (Theorem 2.2 in Section 2). A sketch of proof of Theorem 2.2 is presented in Section 3. For the sake of completeness, a full proof given in [6] is reproduced in Appendix. In Section 5 we shall take up a Schr¨ odinger model with a dissipative term of rank one and obtain Parseval’s formula by the method developed in Sections 2 and 3. In Section 4 we review some results related to ours. Definition 1.1. Let H be a Hilbert space with the inner product (·, ·). Let A be a densely defined linear closed operator in H. A is a dissipative operator if and only if for any u ∈ D(A) Im(Au, u) ≤ 0. The first and second author are supported by Grant-in-Aid for Scientific Research (No.16540161 and No.17740082), Ministry of Education, Culture, Sports, Science and Technology, Japan, respectively.

242

M. Kadowaki, H. Nakazawa and K. Watanabe

We remark that the dissipative operator defined by Lax and Phillips [3] is given by Im(Au, u) ≥ 0. If A is a maximal dissipative operator (i.e., A has no dissipative extension), it is well known that σ(A) (the spectrum of A) is in C− 0 = {z ∈ C; Imz ≤ 0}. A self-adjoint operator is a maximal dissipative operator. We consider the situation that A0 is an absolutely continuous self-adjoint operator with σ(A0 ) = σac (A0 ) = R and V is a bounded dissipative operator. We consider the operator A = A0 + V . 1. Selfadjoint case. (cf. [17], [18]) We first make a review on the selfadjonit case. So, V is self-adjoint and small in some sense. Then the following results are known: the wave operator W− := s- lim eitA0 e−itA Pac (A) t→∞

exists and Range(W− ) = Pac (A0 )H, where Pac (A0 ) is the projection onto the absolutely continuous subspace of A0 , in our case coinciding with H. Furthermore, we know that Ker(W− ) = Ppp (A)H, where Ppp (A) is the projection onto the subspace of the pure point of A, and that A has no singular continuous spectrum. Therefore H = Ppp (A)H ⊕ Pac (A)H (orthogonal sum). As a consequence of these results we have that e−itA u (u = 0) is asymptotically free as t → ∞, i.e., there exists u0 ∈ H such that e−itA u ∼ e−itA0 u0 , t → ∞, if and only if u ∈ / kerW− . 2. Dissipative case. Next we assume that V is dissipative. By a similar way to the selfadjoint case, we can define the wave operator by using the evolution semigroup {e−itA }t≥0 .(see [18]). However we do not know the property of the spectrum of A exactly even if V is very small in some sense. Naturally, if λ ∈ σpp (A), Imλ < 0 and Au = λu (u = 0), we know that e−itA u → 0 as t → ∞. But is the converse true? More precisely, if e−itA u → 0 as t → ∞, u = 0, can we conclude that u belongs to the span of all eigenspaces corresponding to non-real eigenvalues? Our approach in the present paper follows the Kato smooth perturbation method (cf. [8]). We consider that V is of rank one, i.e., V = α(·, ϕ)ϕ, α ∈ C, Imα < 0, ϕ ∈ H. Then using the second resolvent equation we see that (A − z)−1 u = (A0 − z)−1 u − α

((A0 − z)−1 u, ϕ) (A0 − z)−1 ϕ. 1 + α((A0 − z)−1 ϕ, ϕ)

As for the singularity of the resolvent of A, for Imz < 0, since (A0 −z)−1 is analytic in Imz < 0, we can neglect the first term of the right-hand side. So the highest singularity w.r.t. z arises from Γ (z) = 1 + α((A0 − z)−1 ϕ, ϕ) = 0 in Imz < 0. And for Imz = 0, since we know that (A0 − z)−1 ≤

1 , |Imz|

The Rank One Dissipative Operator and the Parseval Formula

243

the problem is the boundary value of Γ(λ − i0), λ ∈ R. If A is self-adjoint, we know that 1 (A − z)−1 ≤ . |Imz|

So the zeros of Γ (z) is, at most, of the degree one. On the other hand, if A is dissipative, we do not have similar estimate. Hence, if the zeros of Γ (λ − i0) is of the degree one, we can see that a representation formula for the wave operator similar to the selfadjoint case is still valid. So, the difficulty in the dissipative case lies the proof of the fact that the zeros of Γ (λ − i0) is of degree one. This can be done under Assumption (A.2) in Section 2. We give some comments for the related works. First, the scattering theory for the non-selfadjoint operators has been developed by S.N. Naboko in [11] and then in [12]. Some other results on the subject can be found in T. Kako and K. Yajima [7]. One should note that dissipative Schr¨ odinger operators (in particular, rank one dissipative perturbations) have been extensively studied by Pavlov in a series of works dating back to early 70s. The explicit construction of symmetric functional model suggested by Pavlov (e.g., see [15]) has been also instrumental for the approach later developed by Naboko. One should also note that the wave operators in the situation of rank one operator (although in the selfadjoint case) have been constructed in the same terms as ours in the Yafaev’s book [18]. This construction with no significant changes also applies in the dissipative case. As for the Parseval formula, a closely related discussion for the dissipative operators can be found in Pavlov [16].

2. Wave equation in the space 1-dimensional case We consider the wave equation with a dissipative term in one space dimension: ∂t2 u(x, t) + ∂t u, ϕ 0 ϕ(x) − ∂x2 u(x, t) = 0,

(x, t) ∈ R × R+

(2.1)

where ·, · 0 is the usual inner product in L2 (R), and it is assumed that ϕ ∈ L2s (R)(= {ϕ; (1+|x|)s ϕ ∈ L2 }) for some s > 1/2. We regard (2.1) as a perturbation of ∂t2 u(x, t) − ∂x2 u(x, t) = 0, (x, t) ∈ R × R. (2.2)

Put f (t) = as where

t

(u(x, t), ∂t u(x, t)). Then (2.1) and (2.2) can be written respectively ∂t f (t) = −iAf (t) A=i

0 ∂x2

1 −·, ϕ 0 ϕ

Let H be the Hilbert space

∂t f (t) = −iA0 f (t),

and and

A0 = i

0 ∂x2

1 . 0

H = {f = t (f1 , f2 ) ∈ H; f1 ∈ H 1 (R), f2 ∈ L2 (R)},

244


where H s (R) denotes the usual Sobolev space, with the inner product of f = t (f1 , f2 ) and g = t (g1 , g2 ) defined as f, g = (∂x f1 (x)∂x g1 (x) + f2 (x)g2 (x))dx, R

The norm of H is denoted by · . Then we know that with the domain is D(A) = D(A0 ) = {f = t (f1 , f2 ) ∈ H; f1 ∈ H 2 (R), f2 ∈ L2 (R)},

A0 and A are a selfadjoint and a maximal dissipative operator, respectively. We list some notations. ϕ ˆ means the usual Fourier transformation of ϕ. We denote the resolvent of A (resp. A0 ) as R(z) = (A − z)−1 for z ∈ ρ(A),

(resp. R0 (z) = (A0 − z)−1 for z ∈ ρ(A0 )),

where ρ(A) (resp. ρ(A0 )) is the resolvent set of the operator A (resp. A0 ). We express the resolvent of −d2 /dx2 in L2 (R) r0 (z) = (−d2 /dx2 − z 2 )−1 z ∈ C \ R.

We express the solution of (2.1) as follows: f (t) = e−itA f, f (0) = f ∈ H.

(2.3)

We say that fb is the bound mode, if fb is an eigenfunction corresponding to the real eigenvalue of A, and that fs (= 0) is the scattering mode, if fs is asymptotically free, i.e., there exists f0 ∈ D(A0 ), f0 = 0, such that lim e−itA fs − e−itA0 f0 = 0,

t→∞

and that fd (= 0) is the dissipative mode, if e−itA fd is dissipative, i.e., lim e−itA fd = 0.

t→∞

Theorem 2.1 (cf. Mochizuki [10]). Let A0 and A be as above. Then 1. A has no real eigenvalues; 2. W = s − limt→∞ eitA0 e−itA exists as an operator from H to H and W = 0. Theorem 2.1, 1 implies that (2.3) does not have the bound mode. So the following two questions naturally arise: (Q1) Does (2.3) have the dissipative mode? (Q2) Is it true that (2.3) is equal to the linear combination of scattering and dissipative modes which are found? Our aim in the present paper is to answer (Q1) and (Q2). For that purpose we introduce the following assumptions. Assumptions. (A1) (A2)

ϕ(x) ∈ L21+s (R)

and Φ(λ) ≦ Φ(µ)

for some s > 1/2

(0 ≦ µ ≦ λ),

2 2 where Φ(λ) = |ϕ(λ)| ˆ + |ϕ(−λ)| ˆ .


245

Under the Assumptions (A1) and (A2) we show that the answer to (Q1) and (Q2) are affirmative. In order to see that we investigate some properties of A. First, we describe the spectrum of the operator A (Proposition 2.1) without the proof (cf. [6]) and secondly we derive (generalized) Parseval’s formula for the operator A (Theorem 2.2). We recall the limiting absorption principle for r0 (z). Let s > 1/2 and u ∈ L2s (R). Then the resolvent and its extension to the reals can be expressed ±i r0 (λ ± iκ)u(x) = e±i(λ±iκ)|x−y| u(y)dy 2(λ ± iκ) R

for λ ∈ R and κ ≧ 0 with λκ = 0. Let (l, j) = (0, 1) or (1, 0). Then a straightforward calculation based on the above expression implies that, for every λ ∈ R, two limits λl ∂xj r0 (λ ± i0) = lim(λ ± iκ)l ∂xj r0 (λ ± iκ) κ↓0

(2.4)

exist in the uniform operator topology of B(L2s (R), L2−s (R)). We fix some s > 1/2, denote by ·, · 0 the dual coupling between L2s (R) and L2−s (R), and put Γ (z) = 1 − izr0 (z)ϕ, ϕ 0 ,

Γ (λ ± i0) = 1 − iλr0 (λ ± i0)ϕ, ϕ 0

Σ± = {z ∈ C± ; Γ (z) = 0} and Σ0± = {λ ∈ R; Γ (λ ± i0) = 0}.

For z ∈ C \ R and f = t (f1 , f2 ) ∈ H the resolvent R0 (z) can be rewritten as R0 (z)f = t (r0 (z)(zf1 + if2 ), i∂x r0 (z)∂x f1 + zr0 (z)f2 ),

and for z = λ + iκ ∈ C± , κ r−λ 1 ∞ i ∞ Γ (z) = 1 + Φ(r)dr − Φ(r)dr. 2 −∞ (r − λ)2 + κ2 2 −∞ (r − λ)2 + κ2

(2.5)

(2.6)

Remark 2.1. By (2.6) we can easily see that Σ+ = Σ0+ = ∅. Proposition 2.1 (cf.[6]). Assume (A1) and (A2). Then / ∅, if Γ (−i0) ≧ 0, Σ− = if Γ (−i0) < 0 {iκ0 }, for some κ0 < 0 and Σ0−

=

/

∅, {0},

if Γ (−i0) = 0, if Γ (−i0) = 0.

(2.7)

(2.8)

Moreover, for Γ (−i0) < 0 (resp. Γ (−i0) = 0) it holds that dΓ (z)/dz|z=iκ0 = 0 (resp. dΓ (λ − i0)/dλ|λ=0 = 0). In order to describe our main theorem (Parseval’s formula corresponding to A) we need some notations. We put

ir0 (z)ϕ . v(z) = zr0 (z)ϕ

246


Using the second resolvent equation in the case of the rank one perturbation, we see that z ∈ Σ+ (resp. Σ− ) if and only if z ∈ ρ(A) ∩ C+ (resp. z ∈ ρ(A) ∩ C− ) and R(z)f = R0 (z)f +

if, v(z) v(z) Γ (z)

(2.9)

for any f = t (f1 , f2 ) ∈ H. Define operators F0 , F and G as follows: ⎧ 0 1 ⎪ ⎪ t λfˆ1 (λ) + ifˆ2 (λ) λfˆ1 (−λ) + ifˆ2 (−λ) ⎪ √ √ , if λ > 0 , ⎪ ⎨ 2 2 0 1 (F0 f )(λ) = ⎪ t −λfˆ1 (−λ) − ifˆ2 (−λ) −λfˆ1 (λ) − ifˆ2 (λ) ⎪ ⎪ √ √ , if λ < 0, , ⎪ ⎩ 2 2 and

⎧ 0 1 ⎪ 0 if, v(λ − i0) ⎪ ⎪ ⎪ ⎨ (Ff )(λ) = (F0 f )(λ) + Γ (λ + i0) (F0 ϕ )(λ), 0 1 ⎪ 0 if, v(λ − i0) ⎪ ⎪ ⎪ ⎩ (Gf )(λ) = (F0 f )(λ) − Γ (λ − i0) (F0 ϕ )(λ).

We know that F0 is a unitary operator from H to L2 (R, C2 ).

Remark 2.2. The operator F has the following property: ∞ ∞ λ(Ff )(λ), g˜(λ) C2 dλ. (FAf )(λ), g˜(λ) C2 dλ =

(2.10)

−∞

−∞

for any f ∈ D(A) and g˜ ∈ L2 (R; C2 ). Theorem 2.2. (Parseval formula) Assume (A1) and (A2). Let P be the projection Pf =

−1 f, v(−iκ0 ) v(iκ0 ), f ∈ H, 2π Γ ′ (iκ0 )

and let E = {g ∈ S(R) × S(R); v(−i0), g = 0}.

1. If Γ (−i0) = 0, it holds that ⎧ ∞ ⎪ ⎪ (Ff )(λ), (Gg)(λ) C2 dλ, if Γ (−i0) > 0 ⎪ ⎪ ⎨ −∞ f, g = ∞ ⎪ ⎪ ⎪ ⎪ ⎩ (Ff )(λ), (Gg)(λ) C2 dλ + P f, g , if Γ (−i0) < 0 −∞

for any f, g ∈ H. 2. If Γ (−i0) = 0, it holds that ∞ f, g = (Ff )(λ), (Gg)(λ) C2 dλ, f ∈ H, g ∈ E −∞


247

Remark 2.3. Gilliam and Schulenberger [2] treated Maxwell’s equations with dissipative boundary conditions in half-space in R3 , but they did not deal with perturbed systems of selfadjoint operators. Now we can answer (Q1) and (Q2). By Theorem 2.2, we see that / {0} if Γ (−i0) ≥ 0, KerW = RangeP if Γ (−i0) < 0.

(2.11)

If Γ (−i0) ≧ 0, then any f ∈ H, f = 0, is the scattering mode. In fact, if f = 0, then f0 := W f = 0 and lim e−itA f − e−itA0 f0 = 0.

t→∞

If Γ (−i0) < 0, any f ∈ H is a linear combination of the scattering and the dissipative modes. In fact, f can be decomposed into f = (f − P f ) + P f ≡ fs + fd , where fs (resp. fd ) is the scattering (resp. dissipative) mode.

3. Outline of the proof of Theorem 2.2 We give a sketch of a proof. The most difficulty is the proof of Lemma 3.2 in step 4 will be given in the Appendix. Step 1. Rewrite the time-dependent representation of the wave operator W as follows: (Kato [8]) κ ∞ W f, g = lim R(λ + iκ)f, R0 (λ + iκ)g dλ. κ↓0 π −∞ Using (2.9) (the representation of the resolvent of A), and the definition of F0 and F, we see that ∞ (Ff )(λ), (F0 g)(λ) C2 dλ. W f, g = −∞

Step 2. We put

Hs = {f =t (f1 , f2 );

R

(1 + |x|2 )s (|∂x f1 (x)|2 + |f2 (x)|2 )dx < ∞}.

By (2.8) in Proposition 2.1 we can prove the following formula: for s > 1/2, for f, g ∈ Hs and λ = 0 (Ff )(λ), (Gg)(λ) C2

1 f, v(λ − i0) v(λ + i0), g 2π Γ (λ + i0) 1 f, v(λ + i0) v(λ − i0), g . − 2π Γ (λ − i0) =(F0 f )(λ), (F0 g)(λ) C2 +

(3.1)

248


Step 3. Using the unitarity of F0 : H −→ L2 (R, C2 ), we see that ∞ (Ff )(λ), (Gg)(λ) C2 dλ −∞ ∞ 1 f, v(λ − i0) v(λ + i0), g =f, g + dλ 2π −∞ Γ (λ + i0) ∞ f, v(λ + i0) v(λ − i0), g 1 dλ. − 2π −∞ Γ (λ − i0)

(3.2)

Step 4. In order to calculate the integrals in the right-hand side of (3.2) we use the following two lemmas: Lemma 3.1 (cf. Lemma 2.4 in [6]). Assume (A1), (A2) and Γ (−i0) = 0. Then it holds that 1. Gg belongs to L2 (R; C2 ) for g ∈ E. 2. E is dense in H. Lemma 3.2. For any f ∈ S(R) × S(R) (S(R) is the set of the rapidly decreasing functions), and for any / S(R) × S(R), if Γ (−i0) = 0, g∈ E, if Γ (−i0) = 0, we have and

∞ −∞

1 f, v(λ + i0) v(λ − i0), g dλ = 2π Γ (λ − i0)

∞ −∞

/

0, if Γ (−i0) ≧ 0, P f, g , if Γ (−i0) < 0,

f, v(λ − i0) v(λ + i0), g dλ = 0. Γ (λ + i0)

(3.3)

(3.4)

In order to justify formulas (3.3) and (3.4), we need to show that the pole of the integrands lying in the real axis has degree not greater than 1. Actually, by Proposition 2.1 the integrands satisfy such a condition for f ∈ S(R) × S(R) and g ∈ S(R) × S(R) (resp. g ∈ E) in the case of Γ (−i0) = 0 (resp. Γ (−i0) = 0). Using Lemma 3.1 and 3.2 in (3.2), we finish the proof of Theorem 2.2.

4. Some other results 1. Solvable model: (cf. [1], [4], [14],[15]) ⎧ ∂2 ⎨ i∂t u(x, t) = − 2 u(x, t) + αδ(·), u(·, t) δ(x), ∂x ⎩u(x, 0) = f (x),

(4.1)


249

where α ∈ C, (Imα ≤ 0). We put H0 = −

d2 , dx2

Hα = H0 + α·, δ δ

(Imα ≤ 0). This operator includes a singular perturbation, and we refer to [1] for the basic information. The operator Hα is given by D(Hα ) = U = u + aH0 (H02 + 1)−1 δ | u ∈ H 2 (R), a ∈ C, u, δ = −a(α−1 + δ, H0 (H02 + 1)−1 δ ) , Hα U = H0 u − a(H02 + 1)−1 ϕ, U ∈ D(Hα )

We can rewrite the equation (4.1) as

iut (t) = Hα u(t),

u(0) = f.

Let W (α) = s- lim eitH0 e−itHα . t→∞

The following Theorems 4.1, 4.2 and 4.3 can be found in [4]. Theorem 4.1. (Spectral structure of Hα ) Let α = α1 + iα2 with α1 ≦ 0, α2 ≦ 0. Then the spectrum of Hα is given by / 2 [0, ∞) ∪ {− α4 }, if α1 < 0, σ(Hα ) = [0, ∞), if α1 = 0. Exact classification of the spectrum σ(Hα ) is σess (Hα ) = σc (Hα ) = [0, ∞), and σp (Hα ) =

/

σr (Hα ) = ∅

2

σd (Hα ) = {− α4 }, ∅, if α1 = 0.

if α1 < 0, 2

Moreover the projection associated to the eigenvalue − α4 P−α2 /4 f = −α/2f, e

(α|·|)/2

e

(α|x|)/2

(α1 = 0) is given by .

Theorem 4.2. (i) Assume that α1 < 0 and α2 < 0. Then KerW (α) = RangeP− α2 . 4

(ii) Assume that α1 = 0 and α2 < 0. Then KerW (iα2 ) = {0}. This theorem follows from the Parseval formula given in the following Theorem 4.3. We define Fα = F0 W (α) where F0 is the usual Fourier transformation.

250


Theorem 4.3. (1) (α1 = 0) For any f, g ∈ H ∩L1 (R1 ) and α ∈ {α = α1 + iα2 ; α1 < 0} ≡ D we have α ¯ e(α|·|)/2 , g . Fα f, Fα¯ g = f, g + f, e(α|·|)/2 2 (2) (α1 = 0) iα2 iα2 iα2 lim Fiα2 f, χε F−iα2 g = f, g + e 2 |y| g(y)dy, e 2 |x| f (x)dx ε→0 4 R1 R1

where χa is the characteristic function of {k ∈ R; a ≦ ||k| + α2 /2|} for a > 0. 2. Spectrum of the dissipative operator: (cf. [5]) / wtt (x, t) − Δw(x, t) + b(x)wt (x, t) = 0, (t, x) ∈ (0, ∞) × RN , w(x, 0) = w0 (x), wt (x, 0) = w1 (x),

(4.2)

where N ≥ 1, b(·) ∈ C 1 (RN \ {0}) is a non-negative function. Putting

0 1 w(·, t) , , w(t) = Hb = i Δ −b wt (·, t)

we can rewrite the above equation as

iw t (t) = Hb w(t),

w(0) =

w0 (·) . w1 (·)

Theorem 4.4. Suppose that the coefficient b(x) is given by / if N = 1, 2, (3 − N )|x|−1 , b0 (x) = (N − 1)|x|−1 , if N ≥ 3, and let the initial data w0 and w1 of (4.2) be / |x|f (|x|), if N = 1, , w1 (x) = ∂|x| {w0 (|x|)} . w0 (x) ≡ f (|x|), if N ≥ 2 where f (|x|) = eβ|x| g(|x|), β < 0, g ∈ S ′ (S ′ is of the class of tempered distributions). Then (4.2) has an explicit radial solution given by / |x|f (|x| + t), if N = 1, w(t, x) = f (|x| + t), if N ≥ 2. Therefore if f ∈ B (B is the set of all the bounded, infinitely differentiable functions) then the total energy decays exponentially as t tends to infinity. Theorem 4.5. Assume N ≥ 3 and |b(x)| ≤ b1 |x|−1 in RN for some b1 ∈ (0, N − 2). Then the following inclusion relation holds: ? b21 2 . α σp (Hb ) ⊂ κ = α + iβ ∈ C | β 2 ≤ 2 (N − 2)2 − b1


251

5. Schr¨ odinger model We consider a 1-dimensional Schr¨ odinger operator with rank 1 perturbation: d2 , dx2 where α ∈ C, Imα < 0 and (·, ·) is the usual L2 (R)-inner product. We confine ourselves to the case ϕ(x) = χ[0,1] (x) (characteristic function of the interval [0, 1]) and calculate Γ (z) = 1 + α((A0 − z)−1 ϕ, ϕ). For Imz = 0 using √ 1 √ (A0 − z)−1 f (x) = ei z|x−y| f (y)dy −2i z R we have 1 1 ((A0 − z)−1 ϕ, ϕ) = 2 − 3 (ew − 1) w w √ √ where we put w = i z and for Imz > 0 we take Im z > 0. The following two lemmas are obtained by explicit calculations: A = A0 + α(·, ϕ)ϕ, A0 = −

Lemma 5.1. Let z = −iε, ε > 0. Then we have √ c −i − 3/2 (e−c ε − 1) ((A0 + iε)−1 ϕ, ϕ) = ε ε where c = eiπ/4 . Moreover, lim((A0 + iε)−1 ϕ, ϕ) = 0. ε↓0

Lemma 5.2. Let z = λ − i0, λ > 0. Then we have √ √ 1 i 1 ((A0 − λ + i0)−1 ϕ, ϕ) = − + 3/2 sin λ + 3/2 (cos λ − 1). λ λ λ Moreover, lim((A0 − λ + i0)−1 ϕ, ϕ) = 0. λ↓0

By Lemmas 5.1 and 5.2 we see that Γ (0) = 0. √Next we consider the degree of the zeros of Γ (λ − i0). For simplicity we put x = λ > 0. Theorem 5.1. There is no point x > 0 such that Γ (x2 − i0) = Γ ′ (x2 − i0) = 0. We put α = a + bi, a, b ∈ R, a = 0 and b < 0. We notice that for x = x0 / 1 + a(−x−2 + x−3 sin x) − bx−3 (cos x − 1) = 0, Γ (x2 − i0) = 0 ⇐⇒ ax3 (cos x − 1) + b(−x−2 + x−3 sin x) = 0.

We put f (x) = x − sin x. If Γ (x2 − i0) = Γ ′ (x2 − i0) = 0 we see that / x30 + af (x0 ) − bf ′ (x0 ) = 0, Γ (x20 − i0) = 0 ⇐⇒ af ′ (x0 ) + bf (x0 ) = 0,

(5.1)

252


and Γ ′ (x20 − i0) = 0 ⇐⇒

/

3x20 + af ′ (x0 ) − bf ′′ (x0 ) = 0, af ′′ (x0 ) + bf ′ (x0 ) = 0.

(5.2)

By (5.1) we obtain f ′ (x0 ) =

b x3 , a2 + b 2 0

f (x0 ) = −

a x3 , a2 + b 2 0

(5.3)

and by (5.2) 3b 3a x2 , f ′′ (x0 ) = 2 x2 . (5.4) a2 + b 2 0 a + b2 0 Hence combining (5.3) and (5.4) for f ′ (x0 ) we have x0 = −3a/b. On the other ′ hand, substituting f ′′ (x) = sin x = 3bx2 /(a2 + b2 ) and cos x = 1 − f (x) = 1 + 3ax2 /(a2 + b2 ) to 1 = sin2 x0 + cos2 x0 , we have x0 = −2a/3. If a > 0, −2a/3 is not real. If a < 0, x0 = −3a/b < 0. It implies a contradiction. (In the case a = 0 the proof is simpler.) f ′ (x0 ) = −

6. Appendix We use the following two lemmas and prove Lemma 3.2 given in Section 3. We omit the proofs of Lemmas 6.1 and 6.2. Lemma 6.1. Let s > 1/2 and g ∈ S(R) × S(R). Then one has for z ∈ C, / O(|z|−1 ) (|z| → ∞), v(z), g = O(1) (|z| → 0). Lemma 6.2. It holds that inf ReΓ (z) ≧ 1. Imz≧0

There exists a positive constant C such that lim inf

|z|→∞,Imz≦0

|ReΓ (z)| ≧ C.

We give a proof of (3.3). Let X1 , X2 and Y be some large positive numbers and ε a small positive number. Put F (z) = −

1 f, v(z) v(z), g 2π Γ (z)

for z ∈ C− . In the case of Γ (−i0) < 0, we see by Proposition 2.1 that iκ0 is a simple pole of F (z). According to [17] Theorem XII.5, the projection P corresponding to the eigenvalue iκ0 is given by: −1 P f, g = R(z)f, g dz, f, g ∈ H (6.1) 2πi γ


253

where γ (⊂ C− ) is a contour around iκ0 . By the residue theorem we know that P f, g =

−1 f, v(−iκ0 ) v(iκ0 ), g , f, g ∈ H. 2π Γ ′ (iκ0 )

Then it follows from Proposition 2.1 and Lemmas 6.1, 6.2 that / 4 0, if Γ (−i0) ≧ 0, F (z)dz = −P f, g , if Γ (−i0) < 0, Γ j j=1 where Γ1 : z = λ − iε (λ = X2 → −X1 ),

Γ2 : z = −X1 + iκ (κ = −ε → −Y ),

Γ3 : z = λ − iY

(λ = −X1 → X2 )

and Γ4 : z = X2 + iκ (κ = −Y → −ε).

Let us estimate

Ij =

F (z)dz

(j = 2, 3, 4).

Γj

Note that I2 = I3 =

and I4 =

Y

F (−X1 + iκ)(−i)dκ, ε X2 −X1

F (λ − iY )dλ

ε

F (X2 + iκ)(−i)dκ.

Y

Using Proposition 2.1 we have that Y 1 C |I2 | ≦ C tan−1 dκ = 2 2 X1 ε X1 + κ X2 1 C |I3 | ≦ C tan−1 dλ = 2 2 Y −X1 λ + Y and |I4 | ≦ C

ε

Y

X22

1 C dκ = + κ2 X2

ε Y − tan−1 X1 X1 X2 X1 + tan−1 Y Y

≦

C π , X1 2

≦

C π Y

ε Y C π , tan−1 ≦ − tan−1 X2 X2 X2 2

where C = C(f, g, ϕ) > 0. Noting that −X1 I1 = F (λ − iε)dλ = − X2

X2

−X1

F (λ − iε)dλ,

254


we have C(

1 1 1 + + )≧ X1 X2 Y

/

| − I1 |, if Γ (−i0) ≧ 0 | − I1 − P f, g |, if Γ (−i0) < 0.

Thus taking X1 , X2 , Y → ∞, we have / ∞ F (λ − iε)dλ = −∞

0, if Γ (−i0) ≧ 0, P f, g , if Γ (−i0) < 0.

(6.2)

In the case Γ (−i0) = 0, the following limit exists by L’Hospital theorem and Γ ′ (−i0) = 0 (cf. Proposition 2.1), lim

λ→0

v(λ − i0), g , Γ (λ − i0)

g ∈ E.

Hence, for each λ, the following limit exists: lim F (λ − iε) = −

ε→0

1 f, v(λ + i0) v(λ − i0), g . 2π Γ (λ − i0)

(6.3)

By keeping (2.7) in mind we see by Lemma 6.1 and 6.2 that for sufficiently small ε > 0 there exist positive numbers C1 and C2 independent of ε such that ⎧ ⎨ C1 (|λ| > 1), |F (λ − iε)| ≦ λ2 ⎩ C (|λ| ≦ 1). 2

Thus by (6.2), (6.3) and by Lebesgue’s convergence theorem, we have (3.3). Next we give a sketch of a proof of (3.4). Put G(z) =

1 f, v(z) v(z), g 2π Γ (z)

for z ∈ C+ . Then it follows from Lemma 6.1 and Lemma 6.2 that 4 G(z)dz = 0, j=1

γ ˜j

where

γ˜1 : z = λ + iε (λ = −X1 → X2 ), γ˜2 : z = X2 + iκ (κ = ε → Y ),

γ˜3 : z = λ + iY and

(λ = X2 → −X1 )

γ˜4 : z = −X1 + iκ (κ = Y → ε). By the same argument as in the proof of 1, we have (3.4). Acknowledgement The authors thank the referee for telling us historical remarks and many articles related to this area and for pointing mistakes.


255

References [1] S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Math. Soc. Lect. Note Ser. No. 271, Cambridge Univ. Press, 2000 [2] D.S. Gilliam and J.R. Schulenberger, Electromagnetic waves in a three-dimensional half-space with a dissipative boundary, J. Math. Anal. Appl. 89, 1982, pp. 129–185 [3] P.D. Lax and R.S. Phillips, Scattering Theory, Academic Press, 1967 [4] M. Kadowaki, H. Nakazawa and K. Watanabe, On the asymptotics of solutions for some Schr¨ odinger equations with dissipative perturbations of rank one, Hiroshima Math. J., 34, 2004, pp. 345–369 [5] M. Kadowaki, H. Nakazawa and K. Watanabe, Exponential decay and spectral structure for wave equation with some dissipations, Tokyo J. Math, 28, 2005, pp. 463–470 [6] M. Kadowaki, H. Nakazawa and K. Watanabe, Parseval formula for wave equations with dissipative term of rank one, to appear in SUT J. Math. [7] T. Kako and K. Yajima, Spectral and scattering theory for a class of non-selfadjoint operators, Sci. Papers College Gen. Ed. Univ. Tokyo 26 1976, pp. 73–89 [8] T. Kato, Wave operators and similarity for some non-self adjoint operators, Math. Ann. 162, 1966, pp. 258–279 [9] S.T. Kuroda, An Introduction to Scattering Theory, Lecture Note Series N o 51, Matematisk Institut, Aarhus University, 1980 [10] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. RIMS Kyoto Univ., 12, 1976, pp. 383–390 [11] S.N. Naboko, A functional model of perturbation theory and its applications to scattering theory , Proc. Steklov Inst. Math., 147, 1980, pp. 86–114 [12] S.N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, Amer. Math. Soc. Transl. Ser. 2, 157, Amer. Math. Soc., Providence, RI, 1993, pp. 127–149 [13] M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunction of a non-selfadjoint operator of second order on a semi axis, Amer. Math. Soc. Transl. Ser, 216, 1960, pp. 103–193 [14] B.S. Pavlov, The nonself–adjoint Schr¨ odinger operators, Topics in Math. Phys. 1, Consultants Bureau New York, 1967, pp. 87–114 [15] B.S. Pavlov, Selfadjoint dilation of a dissipative Schr¨ odinger operator, and expansion in its eigenfunction, (Russian) Math. USSR-Sb. 31, 1977 pp. 457–478 [16] B.S. Pavlov, A remark on spectral meaning of the symmetric functional model, Oper. Theory Adv. Appl., 154, Birkh¨ auser, Basel, 2004, pp. 163–177 [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of Operators, Academic Press, 1978 [18] D.R. Yafaev, Mathematical Scattering Theory: General Theory (Translations of Mathematical Monographs), Amer. Math. Soc., 1992

256


Mitsuteru Kadowaki Department of Mechanical Engineering Ehime University 3 Bunkyo–cho Matsuyama Ehime 790-8577, Japan e-mail: [email protected] Hideo Nakazawa Department of Mathematics Chiba Institute of Technology 2–1–1 Shibazono, Narashino Chiba 275-0023, Japan e-mail: [email protected] Kazuo Watanabe Department of Mathematics Gakushuin University 1–5–1 Mejiro Toshima Tokyo 171-8588, Japan e-mail: [email protected]


On the Schr¨ odinger Operator with Limit-periodic Potential in Dimension Two Yulia Karpeshina and Young-Ran Lee Dedicated to Boris S. Pavlov on the occasion of his 70th birthday.

Abstract. This is an announcement of the following results. We consider the Schr¨ odinger operator H = −Δ + V (x) in dimension two, V (x) being a limitperiodic potential. We prove that the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves eik,x at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous. Mathematics Subject Classification (2000). Primary 35J10; Secondary 81Q15. Keywords. Schr¨ odinger, limit-periodic potential.

1. Main results We study the operator H = −∆ + V (x) in two dimensions,V (x) being a limit-periodic potential: ∞ V (x) = Vr (x),

(1.1)

(1.2)

r=1

where {Vr }∞ r=1 is a family of periodic potentials with doubling periods and decreasing L∞ -norms, namely, Vr has orthogonal periods 2r−1 β1 , 2r−1 β2 and Vr ∞ < ηr ˆ ) for some η > η0 > 0. Without loss of generality we assume that Cexp(−2 Cˆ = 1 and Qr Vr (x)dx = 0, Qr being the elementary cell of periods corresponding to Vr (x). Research partially supported by USNSF Grant DMS-0800949.

258

Y. Karpeshina and Y.-R. Lee

The one-dimensional analog of (1.1), (1.2) is already thoroughly investigated. It is proven in [1]–[7] that the spectrum of the operator H1 u = −u′′ + V u is generically a Cantor type set. It has a positive Lebesgue measure [1, 6]. The spectrum is absolutely continuous [1, 2], [5]–[9]. Generalized eigenfunctions can be represented in the form of eikx u(x), u(x) being limit-periodic [5, 6, 7]. The case of a complex-valued potential is studied in [10]. Integrated density of states is investigated in [11]–[14]. Properties of eigenfunctions of the discrete multidimensional limit-periodic Schr¨ odinger operator are studied in [15]. As to the continuum multidimensional case, it is proven [14] that the integrated density of states for (1.1) is the limit of densities of states for periodic operators. We concentrate here on properties of the spectrum and eigenfunctions of (1.1), (1.2) in the high energy region. We prove the following results for the twodimensional case. 1. The spectrum of the operator (1.1), (1.2) contains a semiaxis. A proof of an analogous result by different means can be found in in the paper [16]. In [16], the authors consider the operator H = (−∆)l + V , 8l > d + 3, d = 1(mod4), d being the dimension of the space. This obviously includes our case l = 1, d = 2. However, there is an additional rather strong restrictions on the potential V (x) in [16], which we don’t have here: in [16] all the lattices of periods Qr of periodic potentials Vr need to contain a nonzero vector γ in common, i.e., V(x) is periodic in a direction γ. 2. There are generalized eigenfunctions Ψ∞ (k, x), corresponding to the semiaxis, which are close to plane waves: for every k in an extensive subset G∞ of R2 , there is a solution Ψ∞ (k, x) of the equation HΨ∞ = λ∞ Ψ∞ which can be described by the formula: . (1.3) Ψ∞ (k, x) = eik,x 1 + u∞ (k, x) , u∞ =|k|→∞ O(|k|−γ1 ),

γ1 > 0,

(1.4)

where u∞ (k, x) is a limit-periodic function: u∞ (k, x) =

∞

ur (k, x),

(1.5)

r=1

ur (k, x) being periodic with periods 2r−1 β1 , 2r−1 β2 . The eigenvalue λ∞ (k) corresponding to Ψ∞ (k, x) is close to |k|2 : λ∞ (k) =|k|→∞ |k|2 + O(|k|−γ2 ),

γ2 > 0.

(1.6)

The “non-resonant” set G∞ of the vectors k, for which (1.3)–(1.6) hold, is ∞ an extensive Cantor type set: G∞ = ∩∞ n=1 Gn , where {Gn }n=1 is a decreasing sequence of sets in R2 . Each Gn has a finite number of holes in each bounded region. More and more holes appear when n increases, however holes added

The Schr¨ odinger Operator with Limit-periodic Potential

259

at each step are of smaller and smaller size. The set G∞ satisfies the estimate: |(G∞ ∩ BR )| =R→∞ 1 + O(R−γ3 ), |BR |

γ3 > 0,

(1.7)

where BR is the disk of radius R centered at the origin, | · | is the Lebesgue measure in R2 . 3. The set D∞ (λ), defined as a level (isoenergetic) set for λ∞ (k), D∞ (λ) = k ∈ G∞ : λ∞ (k) = λ ,

is proven to be a slightly distorted circle with infinite number of holes. It can be described by the formula: D∞ (λ) = {k : k = κ∞ (λ, ν )ν , ν ∈ B∞ (λ)}, (1.8)

where B∞ (λ) is a subset of the unit circle S1 . The set B∞ (λ) can be interpreted as the set of possible directions of propagation for the almost plane waves (1.3). The set B∞ (λ) has a Cantor type structure and an asymptotically full measure on S1 as λ → ∞: . L (Bn (λ)) =λ→∞ 2π + O λ−γ3 /2 , (1.9)

here and below L(·) is the length of a curve. The value κ∞ (λ, ν ) in (1.8) is the “radius” of D∞ (λ) in a direction ν . The function κ∞ (λ, ν ) − λ1/2 describes the deviation of D∞ (λ) from the perfect circle of the radius λ1/2 . It is proven that the deviation is small: κ∞ (λ, ν ) =λ→∞ λ1/2 + O λ−γ4 , γ4 > 0. (1.10)

4. Absolute continuity of the branch of the spectrum (the semiaxis) corresponding to Ψ∞ (k, x) is proven. To prove the results listed above we develop a modification of the KolmogorovArnold-Moser (KAM) method. This paper is inspired by [17, 18, 19], where the method is used for periodic problems. In [17] KAM method is applied to classical Hamiltonian systems. In [18, 19] the technique developed in [17] is applied to semiclassical approximation for multidimensional periodic Schr¨ odinger operators at high energies. We consider a sequence of operators H0 = −∆,

H (n) = H0 +

Mn r=1

Vr ,

n ≥ 1, Mn → ∞ as n → ∞.

Obviously, H − H (n) → 0 as n → ∞ and H (n) = H (n−1) + Wn where Wn = Mn (n) , n ≥ 1, as a perturbation of the r=Mn−1 +1 Vr . We consider each operator H previous operator H (n−1) . Every operator H (n) is periodic, however the periods go to infinity as n → ∞. We show that there is a λ∗ , λ∗ = λ∗ (V ), such that the semiaxis [λ∗ , ∞) is contained in the spectra of all operators H (n) . For every operator H (n) there is a set of eigenfunctions (corresponding to the semiaxis) being

260

Figure 1


Distorted circle with holes, D1 (λ)

Figure 2

Distorted circle with holes, D2 (λ)

close to plane waves: for every k in an extensive subset Gn of R2 , there is a solution Ψn (k, x) of the differential equation H (n) Ψn = λn Ψn , which can be described by the formula: . Ψn (k, x) = eik,x 1 + u ñ (k, x) , ˜ un L (R2 ) = O(|k|−γ1 ), γ1 > 0, (1.11) ∞

| k|→∞

1 , 2Mn −1 β 2 .1 The corresponding eigenvalue where u ñ (k, x) has periods 2Mn −1 β λ(n) (k) is close to |k|2 : . λ(n) (k) =|k|→∞ |k|2 + O |k|−γ2 , γ2 > 0. The non-resonant set Gn is proven to be extensive in R2 : |(G∞ ∩ BR )| =R→∞ 1 + O(R−γ3 ). |BR |

(1.12)

Estimates (1.11)–(1.12) are uniform in n. The set Dn (λ) is defined as the level (isoenergetic) set for non-resonant eigenvalue λ(n) (k): Dn (λ) = k ∈ Gn : λ(n) (k) = λ .

This set is proven to be a slightly distorted circle with a finite number of holes (Fig. 1, 2), the set D1 (λ) being strictly inside the circle of the radius λ1/2 for sufficiently large λ. The set Dn (λ) can be described by the formula: Dn (λ) = {k : k = κn (λ, ν )ν , ν ∈ Bn (λ)}, (1.13) 1 Obviously,

u ñ (k, x) is simply related to functions ur (k, x) used in (1.5): Mn x) = ur (k, x). u ñ (k, r=Mn−1 +1


261

where Bn (λ) is a subset of the unit circle S1 . The set Bn (λ) can be interpreted as the set of possible directions of propagation for almost plane waves (1.11). It has an asymptotically full measure on S1 as λ → ∞: . L (Bn (λ)) =λ→∞ 2π + O λ−γ3 /2 . (1.14)

The set Bn has only a finite number of holes, however their number is growing with n. More and more holes of a smaller and smaller size are added at each step. The value κn (λ, ν ) − λ1/2 gives the deviation of Dn (λ) from the perfect circle of the radius λ1/2 in the direction ν . It is proven that the deviation is asymptotically small: ∂κn (λ, ν ) = O λ−γ5 , γ4 , γ5 > 0, κn (λ, ν ) = λ1/2 + O λ−γ4 , (1.15) ∂ϕ ϕ being an angle variable ν = (cos ϕ, sin ϕ). On each step more and more points are excluded from the non-resonant sets Gn , thus {Gn }∞ n=1 is a decreasing sequence of sets. The set G∞ is defined as the limit set: G∞ = ∩∞ n=1 Gn . It has an infinite number of holes at each bounded region, but nevertheless satisfies the relation (1.7). For every k ∈ G∞ and every n, there is a generalized eigenfunction of H (n) of the type (1.11). It is proven that the sequence of Ψn (k, x) has a limit in L∞ (R2 ) as n → ∞, when k ∈ G∞ . The function Ψ∞ (k, x) = limn→∞ Ψn (k, x) is a generalized eigenfunction of H. It can be written in the form (1.3)–(1.5). Naturally, the corresponding eigenvalue λ∞ (k) is the limit of λ(n) (k) as n → ∞. It is shown that {Bn (λ)}∞ n=1 is a decreasing sequence of sets, since on each step more and more directions are excluded. We consider the limit B∞ (λ) of Bn (λ): B∞ (λ) =

∞ @

n=1

Bn (λ).

This set has a Cantor type structure on the unit circle. It is proven that B∞ (λ) has an asymptotically full measure on the unit circle (see (1.9)). We prove that the sequence κn (λ, ν ), n = 1, 2, . . . , describing the isoenergetic curves Dn , quickly converges as n → ∞. Hence, D∞ (λ) can be described as the limit of Dn (λ) in the sense (1.8), where κ∞ (λ, ν ) = limn→∞ κn (λ, ν ) for every ν ∈ B∞ (λ). It is shown that the derivatives of the functions κn (λ, ν ) (with respect to the angle variable on the unit circle) have a limit as n → ∞ for every ν ∈ B∞ (λ). We denote this (λ, ν) limit by ∂κ∞∂ϕ . Using (1.15) we prove that ∂κ∞ (λ, ν ) = O λ−γ5 . ∂ϕ

Thus, the limit curve D∞ (λ) has a tangent vector in spite of its Cantor type structure, the tangent vector being the limit of the corresponding tangent vectors for Dn (λ) as n → ∞. The curve D∞ (λ) looks as a slightly distorted circle with infinite number of holes.

262


Absolute continuity of the branch of the spectrum [λ∗ (V ), ∞), corresponding to the functions Ψ∞ (k, x), k ∈ G∞ , follows from continuity properties of level curves D∞ (λ) with respect to λ, and from convergence of spectral projections corresponding to Ψn (k, x), k ∈ G∞ , to spectral projections of H in the strong sense and uniformly in λ, λ > λ∗ . The main technical difficulty overcome is construction of non-resonance sets Bn (λ) for every fixed sufficiently large λ, λ > λ0 (V ), where λ0 (V ) is the same for all n. The set Bn (λ) is obtained by deleting a “resonant” part from Bn−1 (λ). Definition of Bn−1 \Bn includes Bloch eigenvalues of H (n−1) . To describe Bn−1 \Bn one has to use not only non-resonant eigenvalues of the type (1.6), but also resonant eigenvalues, for which no suitable formulae are known. Absence of formulae cause difficulties in estimating the size of Bn \ Bn−1 . To deal with this problem we start with introducing an angle variable ϕ, ϕ ∈ [0, 2π), ν = (cos ϕ, sin ϕ) ∈ S1 and consider sets Bn (λ) in terms of this variable. Next, we show that the resonant set B n−1 \ Bn can be described as the set of zeros of determinants of the type Det I + Sn (ϕ) , Sn (ϕ) being a trace type operator, .−1 .- I + Sn (ϕ) = H (n−1) κ , n−1 (ϕ) + b − λ − ǫ H0 κ n−1 (ϕ) + b + λ where κ n−1 (ϕ) is a vector-function describing Dn−1 (λ), κ n−1 (ϕ) = κn−1 (λ, ν )ν . To obtain Bn−1 \ Bn we take all values of ǫ in a small interval and values of b in a finite set, b = 0. Further, we extend our considerations to a complex neighborhood Φ0 of [0, 2π). We show that the determinants are analytic functions of ϕ in Φ0 , and, by this, reduce the problem of estimating the size of the resonance set to a problem in complex analysis. We use theorems for analytic functions to count zeros of the determinants and to investigate how far the zeros move when ε changes. It enables us to estimate the size of the zero set of the determinants, and, hence, the size of the non-resonance set Φn ⊂ Φ0 , which is defined as a non-zero set for the determinants. Proving that the non-resonance set Φn is sufficiently large, we obtain estimates (1.12) for Gn and (1.14) for Bn , the set Bn being the real part of Φn . To obtain Φn we delete from Φ0 more and more holes of smaller and smaller radii at each step. Thus, the non-resonance set Φn ⊂ Φ0 has a structure of Swiss Cheese (Fig. 3, 4). Deleting resonance set from Φ0 at each step of the recurrent procedure is called by us a “Swiss Cheese Method”. The essential difference of our method from those applied in similar situations before (see, e.g., [17]–[20]) is that we construct a non-resonance set not only on the whole space of a parameter (k ∈ R2 here), but also on all isoenergetic curves Dn (λ) in the space of parameter, corresponding to sufficiently large λ. Estimates for the size of non-resonance sets on a curve require more subtle technical considerations than those sufficient for description of a non-resonant set in the whole space of the parameter. The case of the operator H = (−∆)l +V , l ≥ 6, d = 2 is considered in [23]. The results proved in [23] are analogous to 1-4 on pages 258, 259. The restriction l ≥ 6 is technical, it is needed only for the first two steps of the recurrent procedure. We modify the proof in [23] to extend the results to the case l = 1. The requirement for


263

Φ2

Figure 3. The set Φ2 .

Φ3

Figure 4. The set Φ3 . super exponential decay of Vr as r → ∞ is essential, since it is needed to ensure convergence of the recurrent procedure. At every step we use the upper bounds on Vr to prove perturbation formulae for Bloch eigenvalues and eigenfunctions when λ > λ∗ (V ), λ∗ being the same for all steps. Then, these perturbation formulae are used to prove existence of the limits of λ(n) (k), κn (λ, ν ), Ψn (k, x) as n → ∞ and, hence, to define λ∞ (k), κ∞ (λ, ν ), Ψ∞ (k, x). It is not particularly important that potentials Vr have doubling periods, in the sense that the periods of the type 1 , q r−1 β2 , q ∈ N, can be treated in the same way as doubling. q r−1 β

References [1] J. Avron, B. Simon Almost Periodic Schr¨ odinger Operators I: Limit Periodic Potentials. Commun. Math. Physics, 82 (1981), 101–120. [2] V.A. Chulaevski On perturbation of a Schr¨ odinger Operator with Periodic Potential. Russian Math. Surv., 36(5), (1981), 143–144. [3] J. Moser An Example of the Schr¨ odinger Operator with Almost-Periodic Potentials and Nowhere Dense Spectrum. Comment. Math. Helv., 56 (1981), 198–224.

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[4] B. Simon Almost Periodic Schr¨ odinger Operators. A Review. Advances in Applied Mathematics, 3 (1982), 463–490. [5] L.A. Pastur, V.A. Tkachenko On the Spectral Theory of the One-Dimensional Schr¨ odinger Operator with Limit-Periodic Potential. Dokl. Akad. Nauk SSSR, 279 (1984), 1050–1053; Engl. Transl.: Soviet Math. Dokl., 30 (1984), no. 3, 773–776 [6] L.A. Pastur, V.A. Tkachenko Spectral Theory of a Class of One-Dimensional Schr¨ odinger Operators with Limit-Periodic Potentials. Trans. Moscow Math. Soc., 51 (1989), 115–166. [7] L. Pastur, A. Figotin Spectra of Random and Almost-Periodic Operators. SpringerVerlag, 1992, 583pp. [8] J. Avron, B. Simon Cantor Sets and Schr¨ odinger Operators: Transient and Recurrent Spectrum. J. Func. Anal., 43 (1981), 1–31. [9] S.A. Molchanov and V.A. Chulaevskii Structure of the Spectrum of Lacunary LimitPeriodic Schr¨ odinger Operator. Func. Anal. Appl., 18 (1984), 91–92. [10] L. Zelenko On a Generic Topological Structure of the Spectrum to One-Dimensional Schr¨ odinger Operators with Complex Limit-Periodic Potentials. Integral Equations and Operator Theory, 50 (2004), 393–430. [11] M.A. Shubin The Density of States for Selfadjoint Elliptic Operators with Almost Periodic Coefficients. Trudy sem. Petrovskii (Moscow University), 3 (1978), 243–275. [12] M.A. Shubin Spectral Theory and Index of Elliptic Operators with Almost Periodic Coefficients. Russ. Math. Surveys, 34(2), (1979), 109–157. [13] J. Avron, B. Simon Almost Periodic Schr¨ odinger Operators. II: The Integrated Density of States. Duke Math. J., 50 (1983), 1, 369–391. [14] G.V. Rozenblum, M.A. Shubin, M.Z. Solomyak Spectral Theory of Differential Operators. Encyclopaedia of Mathematical Sciences, 64, 1994. [15] Yu.P. Chuburin On the Multidimensional Discrete Schr¨ odinger Equation with a Limit Periodic Potential. Theoretical and Mathematical Physics, 102 (1995), no. 1, 53–59. [16] M.M. Skriganov, A.V. Sobolev On the spectrum of a limit-periodic Schr¨ odinger operator Algebra i Analiz, 17 (2005), 5; English translation: St. Petersburg Math. J. 17 (2006), 815–833. [17] G. Gallavotti, Perturbation Theory for Classical Hamiltonian Systems. Scaling and Self-Similarity in Progr. Phys., 7, edited by J. Froehlich, Birkhäuser, Basel, Switzerland, 1983, 359–424. [18] L.E. Thomas, S.R. Wassel, Stability of Hamiltonian systems at high energy. J. Math. Phys. 33(10), (1992), 3367–3373. [19] L.E. Thomas and S.R. Wassel, Semiclassical Approximation for Schr¨ odinger Operators at High Energy, Lecture Notes in Physics, 403, edited by E. Balslev, SpringerVerlag, 1992, 194–210. [20] J. Bourgain, Quasiperiodic Solutions of Hamilton Perturbations of 2D Linear Schr¨ odinger Equation. Ann. of Math. (2), 148 (1998), 2, 363–439. [21] M. Reed, B. Simon, Methods of Modern Mathematical Physics., Vol IV, Academic Press, 3rd ed., New York – San Francisco – London (1987), 396 pp.


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[22] Yu. Karpeshina Perturbation theory for the Schr¨ odinger operator with a periodic potential. Lecture Notes in Mathematics, 1663, Springer-Verlag, 1997, 352 pp. [23] Yu. Karpeshina, Y.-R. Lee Spectral Properties of Polyharmonic Operators with LimitPeriodic Potential in Dimension Two. D’Analyse Mathématique, 102 (2007), pp. 225–310. Yulia Karpeshina Department of Mathematics University of Alabama at Birmingham 1300 University Boulevard Birmingham, AL, 35294-1170, USA e-mail: [email protected] Young-Ran Lee Department of Mathematical Sciences KAIST (Korea Advanced Institute of Science and Technology) 335 Gwahangno, Yusong-Gu Daejeon, 305-701, Republic of Korea e-mail: [email protected]


Similarity Problem for Non-self-adjoint Extensions of Symmetric Operators Alexander V. Kiselev Abstract. The similarity problem for non-self-adjoint extensions of a symmetric operator having equal deficiency indices is studied. Necessary and sufficient conditions for a wide class of such operators to be similar to self-adjoint ones are obtained. The paper is based on the construction of functional model for the operators of the class considered due to Ryzhov [26, 25] and extends the results of [10] to this class. In addition to solving the similarity problem for the operator itself, we also give necessary and sufficient conditions for similarity of its restrictions to spectral subspaces corresponding to arbitrary Borel sets of the real line. These conditions (together with the technique employed while establishing them) have their direct analogues in the setting of the paper [10]. Mathematics Subject Classification (2000). 47A10, 47A20, 47A45, 47A48, 47A55. Keywords. Extensions, functional model, non-self-adjoint operators, symmetric operators, similarity theory.

1. Introduction A non-self-adjoint operator L acting in the Hilbert space H is called similar to a self-adjoint operator A if there exists a bounded, boundedly invertible operator X in H such, that L = X −1 AX. The similarity problem thus arises, i.e., to ascertain whether a given non-self-adjoint operator L is similar to some self-adjoint operator A or not and to give preferably necessary and sufficient conditions of this. A criterion for similarity of a general non-self-adjoint operator with real spectrum to a self-adjoint one was established in a form of a pair of integral estimates involving the resolvent of the operator in [17, 5] (see (3.1) below). Unfortunately, this criterion, although given in a very concise form, is hard to verify in applications. This is why it seems reasonable to rewrite it in an equivalent (and suitable The author gratefully appreciates financial support of INTAS (grant no. 05-1000008-7883) and RFBR (grant no. 06-01-00249).

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for applications) form under some additional assumptions on the class of operators considered. One such assumption that has yielded rather interesting results in the case of additive non-self-adjoint perturbations of self-adjoint operators (see [10]) is that the spectrum of the operator L is absolutely continuous, i.e., the absolutely continuous spectral subspace of the operator coincides with the Hilbert space H (see Section 2 below for the definition and Remark 3.2). In the present paper we consider the similarity problem for non-self-adjoint extensions of symmetric operators with equal deficiency indices, i.e., we work out the conditions necessary and sufficient for such operators to be similar to selfadjoint ones, extending the results of [10] to the class of non-self-adjoint almost solvable extensions of symmetric operators with absolutely continuous spectrum. Namely, Theorems 3.3 and 3.5 and Corollaries 3.4 and 3.6 below have the same form as the corresponding statements of [10]. The approach employed (based on the functional model of non-self-adjoint operators and quite similar to that of [10]) has also permitted us to study the similarity problem for restrictions of the given non-self-adjoint operator to spectral subspaces of the operator L corresponding to arbitrary Borel sets of the real line. Necessary and sufficient conditions of the named similarity have been obtained. Moreover, direct analogues of these conditions also hold in the setting of the paper [10], although the named paper contains no discussion of this subject. It should be also noted that in [10] the results on the similarity problem have been published without the proof. Results on the similarity problem for restrictions of additive non-self-adjoint perturbations of self-adjoint operators, similar to the ones contained in the present paper, will be published elsewhere. The conditions of similarity to a self-adjoint operator derived in the present paper differ from ones that have been known previously (that is, the ones of [17, 5]) in that they are formulated in terms of objects defined on the real line only, whereas the existing ones were formulated in terms of half-planes of the complex plane. In a nutshell, using the functional model we have been able to pass to a limit in the Naboko-Van Casteren criterion under the only additional assumption that the operator L has absolutely continuous spectrum. Since the functional model of a non-self-adjoint operator is of crucial importance for our approach, the proofs of our main results rely heavily upon the recent paper by Ryzhov [25], where the symmetric form of the Nagy-Foia¸s functional model has been developed for almost solvable non-self-adjoint extensions of symmetric operators due to Pavlov [21, 20] (see also the paper [16] by Naboko, where this idea was first suggested for additive non-self-adjoint perturbations of self-adjoint operators). The approach of Ryzhov is in its turn based on the technique of the so-called boundary spaces. We continue with a brief introduction to the main concepts and results obtained in this area in Section 2. Section 3 contains our main result, which is a criterion of similarity of a nonself-adjoint operator of the class considered to a self-adjoint one. This criterion is formulated in two equivalent forms. Depending on the concrete choice of the operator under consideration, one of these forms might prove to be preferable

Similarity Problem. . .

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to another from the viewpoint of simplicity of calculations. Further, two concise sufficient conditions of similarity are given that seem to be even more suitable for applications. Section 4 deals with the similarity problem for the above-mentioned restrictions of non-self-adjoint operator under investigation to spectral subspaces corresponding to arbitrary Borel sets of the real line. We formulate results that fall in line with those of Section 3 and sketch their proof. Finally, Section 5 demonstrates applicability of our results to analysis of the similarity problem for non-self-adjoint Laplace operator with a potential of zero radius in the case of R3 .

2. The functional model for almost solvable extensions of symmetric operators The narrative of the present section in the part concerning boundary triples and the theory of almost solvable extensions is based on the works [8, 7, 4, 12], in the part concerning explicit construction of Pavlov-Naboko functional model – on the paper [25], see also [16]. Suppose that A0 is a symmetric densely defined closed linear operator acting in the Hilbert space H (D(A0 ) ≡ DA0 and R(A0 ) ≡ RA0 denoting its domain and range respectively; D(A∗0 ) ≡ DA∗0 , R(A∗0 ) ≡ RA∗0 denoting the domain and range of operator A∗0 adjoint to A0 ). Assume that A0 is completely non-self-adjoint, i.e., there exists no subspace H0 in H such that the restriction A0 |H0 is a self-adjoint operator in H0 . Further assume that the deficiency indices of A0 (probably being infinite) are equal: n+ (A0 ) = n− (A0 ) ≤ ∞. Definition 2.1 ([8, 4, 12]). Let Γ1 , Γ2 be linear mappings of DA∗0 to H – a separable Hilbert space. The triple (H, Γ1 , Γ2 ) is called a boundary triple for the operator A∗0 if: 1. for all f, g ∈ DA∗0

(A∗0 f, g)H − (f, A∗0 g)H = (Γ1 f, Γ2 g)H − (Γ2 f, Γ1 g)H .

2. the mapping γ defined as f −→ (Γ1 f ; Γ2 f ), f ∈ DA∗0 is surjective, i.e., for all Y1 , Y2 ∈ H there exists such y ∈ DA∗0 that Γ1 y = Y1 , Γ2 y = Y2 . A boundary triple can be constructed for any operator A0 of the class considered. Moreover, the space H can be chosen in a way such that dim H = n+ = n− . Definition 2.2 ([8, 7]). A nontrivial extension A˜B of the operator A0 such that A0 ⊂ A˜B ⊂ A∗0 is called almost solvable if there exists a boundary triple (H, Γ1 , Γ2 ) for A∗0 and a bounded linear operator B defined everywhere on H such that for every f ∈ DA∗0 f ∈ DA˜B if and only if Γ1 f = BΓ2 f.

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It can be shown that if an extension A˜B of A0 , A0 ⊂ A˜B ⊂ A∗0 , has regular points (i.e., the points belonging to the resolvent set) in both upper and lower half-planes of the complex plane, then this extension is almost solvable. The following theorem holds: Theorem 2.3 ([8, 7]). Let A0 be a closed densely defined symmetric operator with n+ (A0 ) = n− (A0 ), let (H, Γ1 , Γ2 ) be a boundary triple of A∗0 . Consider the almost solvable extension A˜B of A0 and the corresponding bounded operator B in H. Then: 1. 2. 3. 4. 5.

y ∈ DA0 if and only if Γ1 y = Γ2 y = 0, A˜B is maximal, i.e., ρ(A˜B ) = ∅, (A˜B )∗ ⊂ A∗0 , (A˜B )∗ = A˜(B ∗ ) , operator A˜B is dissipative if and only if B is dissipative, (A˜B )∗ = A˜B if and only if B ∗ = B.

Definition 2.4 ([7, 25]). Let A0 be a closed densely defined symmetric operator, n+ (A0 ) = n− (A0 ), (H, Γ1 , Γ2 ) is its space of boundary values. The operatorfunction M (λ), defined by M (λ)Γ2 fλ = Γ1 fλ , fλ ∈ ker(A∗0 − λI), λ ∈ C± ,

(2.1)

is called the Weyl function of a symmetric operator A0 . The following theorem describing the properties of the Weyl function clarifies its meaning. Theorem 2.5 ([7]). Let M (λ) be the Weyl function of the symmetric operator A0 with equal deficiency indices (n+ (A0 ) = n− (A0 ) < ∞). Let A˜B be an almost solvable extension of A0 corresponding to a bounded operator B. Then for every λ ∈ C:

1. M (λ) is analytic operator-function when Im λ = 0, its values being bounded linear operators in H. 2. (Im M (λ)) Im λ > 0 when Im λ = 0. 3. M (λ)∗ = M (λ) when Im λ = 0. 4. λ ∈ σp (A˜B ) if and only if det(M (λ) − B) = 0. 5. λ ∈ ρ(A˜B ) if and only if B − M (λ) is boundedly invertible.

Consider now the definition and the properties of the characteristic operator-function of a non-self-adjoint extension of a closed, densely defined symmetric operator [28, 25]. Let A0 be a symmetric operator, (H, Γ1 , Γ2 ) its boundary triple and M (λ) the corresponding Weyl function. Let A˜B be an almost solvable extension of A0 (we assume for simplicity that it is completely non-self-adjoint). Consider the following definitions: G(A˜B ) ≡ {g ∈ D(A˜B ) : (A˜B f, g) − (f, A˜B g) = 0 for allf ∈ D(A˜B )}, L(A˜B ) ≡ D(A˜B )/G(A˜B ).


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Definition 2.6. ([28, 29]) Hilbert space L isomorphic to the closure of L(A˜B ) w.r.t. the metric [·, ·] induced by the non-degenerate but not necessarily positive inner product (A˜B f1 , f2 ) − (f1 , A˜B f2 ) [ξ1 , ξ2 ]L ≡ , ξ1 , ξ2 ∈ L, fk ∈ ξk , k = 1, 2, 2i is called the boundary space of operator A˜B . The mapping Γ : D(A˜B ) → L, for which (A˜B f1 , f2 ) − (f1 , A˜B f2 ) , f1 , f2 ∈ D(A˜B ), [Γf1 , Γf2 ]L = 2i is called the boundary operator for A˜B . Let L′ , Γ′ be respectively the boundary space and boundary operator for the operator (−A˜B )∗ . Characteristic operator-function ΘA˜B [3] of the operator A˜B is defined by the following equality: ΘA˜B (λ)Γf = Γ′ gλ ,

(2.2)

where λ ∈ ρ(A˜∗B ), f ∈ D(A˜B ), gλ ≡ (A˜∗B − λI)−1 (A˜B − λI)f.1 Considering the polar decomposition of the bounded operator Im B ≡ (B − B ∗ )/2i acting in the auxiliary Hilbert space E ≡ clos(Ran(Im B)), Im B = Jα2 /2, where α ≡ |2 Im B|1/2 , J ≡ sign(Im B|E) it can be proved [25] that the space E with the metric [·, ·] ≡ (J·, ·)H can be chosen as a boundary space for the operator A˜B ; in this case the boundary operator Γ can be defined as follows [12, 8, 7], see also [25]: Γ ≡ JαΓ2 ; D(Γ) ≡ D(A˜B ). and the characteristic function for A˜B is given by the following formula: Θ(λ) ≡ I|E + iJα(B ∗ − M (λ))−1 α|E , λ ∈ ρ(A˜∗B ).

(2.3)

Now consider the functional model for a non-self-adjoint extension of symmetric operator A0 [20, 30, 16, 25]. Alongside with an arbitrary almost solvable extension A˜B consider the dissipative [24, 20] extension A˜B+ , corresponding to the bounded operator B+ defined as follows: B = ℜB + iαJα/2 −→ B+ ≡ ℜB + i| Im B| = ℜB + iα2 /2.

As stated in [25], the requirement of complete non-self-adjointness of the operator A˜B leads to the fact that the dissipative operator A˜B+ is also completely non-selfadjoint. Introduce simplified notation: A ≡ A˜B , A+ ≡ A˜B+ , and further denote the characteristic function of the operator A+ by the symbol S(λ). It is a contractive in the upper half-plane analytic operator function (this can be quite easily verified based on [8], see, e.g., [25]). Thus S(λ) possesses limits almost everywhere on the 1 For

˜∗ ) the operator-function Θ ˜ each λ ∈ ρ(A B A

L′ (see [3]).

B

acts as a linear bounded operator from L onto

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real line in the strong topology, which we denote by S(k). Consider the model ∗ space H = L2 ( SI SI ). It is defined in [20] (see also [18] for description of general coordinate-free models) in the following way: consider all two-component vectorfunctions (˜ g , g) on the axis (˜ g (k), g(k) ∈ E, k ∈ R) finite w.r.t. the following metric:

B

A B ∞ A g˜(k) g˜(k) g˜ g˜ I S ∗ (k) , = , dk. S(k) I g(k) E⊕E g(k) g g −∞ Factoring the set of two-component functions by the set of elements with norm equal to zero and then closing the result w.r.t. the above norm, we finally arrive at the model space H. Although we consider (˜ g , g) as a symbol only, the formal expressions g− := (˜ g + S ∗ g) and g+ := (S˜ g + g) (the motivation for the choice of notation is selfevident from what follows) can be shown to represent some true L2 (E)-functions on the real line. In what follows we plan to deal mostly with these functions. Define the following orthogonal subspaces in H:

2 H+ (E) 0 , K ≡ H ⊖ (D− ⊕ D+ ), , D+ ≡ D− ≡ 2 H− (E) 0

2 where H+(−) (E) denotes the Hardy class [30] of analytic functions f in the upper (lower) half-plane taking values in the Hilbert space E. These subspaces are “incoming” and “outgoing” subspaces, respectively, in the sense of [13]. The subspace K can be described as K = {(˜ g, g) ∈ H : g− ≡ g˜ + S ∗ g ∈ 2 2 H− (E), g+ ≡ S˜ g + g ∈ H+ (E)}. Let PK be the orthogonal projection of the space H onto K, then

g + S ∗ g) g˜ g˜ − P+ (˜ , = PK g − P− (S˜ g + g) g 2 where P± are the orthogonal Riesz projections of the space L2 (E) onto H± (E).

The following theorem holds [25], cf. [16, 20]: Theorem 2.7 (Ryzhov, [25]). The operator (A+ − λ0 )−1 is unitary equivalent to the operator PK (k − λ0 )−1 |K for all λ0 , Im λ0 < 0. This means, that the operator of multiplication by k serves as the minimal (closIm λ =0 (k − λ)−1 K = H) self-adjoint dilation [30] of the operator A+ . As it is shown in [25], cf. [16], it is possible to compute the action of the operator A in the model representation of dissipative operator A+ . In particular, the following theorem holds: Theorem 2.8 (Ryzhov, [25]). Let λ0 ∈ C− ∩ ρ(A); µ0 ∈ C+ ∩ ρ(A); (˜ g , g) ∈ K. Then

g˜ g˜ , = PK (k − λ0 )−1 (A − λ0 I)−1 −1 g + S ∗ g)(λ0 ) g − X− Θ− (λ0 )(˜ g (2.4)

−1 g + g)(µ0 ) g˜ − X+ Θ+ (µ0 )(S˜ g˜ (A − µ0 I)−1 , = PK (k − µ0 )−1 g g


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where X± ≡ (I ± J)/2,

Θ− (λ) ≡ I − iα(B+ − M (λ))−1 αX− = X+ + S ∗ (λ)X− , ∗ Θ+ (λ) ≡ I + iα(B+ − M (λ))−1 αX+ = X− + S(λ)X+ ,

Im λ < 0, Im λ > 0,

and S(λ) is as before the characteristic operator-function of the dissipative operator A+ (see (2.3); [16, 25]). ˆ± in H as follows: Following [15], we define the linear sets N ? g˜ g˜ ˆ± ≡ ∈ H, P± (X+ g+ + X− g− ) = 0 : N g g

(2.5)

ˆ± . and introduce the following designation: N± = clos PK N Absolutely continuous and singular subspaces of the non-self-adjoint operator ˆ+ ∩ N ˆ− , N ˜± ≡ PK N ˆ± , A are defined in the same way as in the paper [14]: let N ≡ N ˜+ ∩ N ˜− . The subspaces N± (A∗ ) for the adjoint to A operator A∗ are defined ˜e ≡ N N in an analogous way using the same model representation. Then . ˜e ; Ni ≡ K ⊖ Ne (A∗ ), ˜+ ∩ N ˜− = clos PK N ≡ clos N Ne ≡ clos N (2.6)

where Ne (A∗ ) denotes the absolutely continuous subspace of the operator A∗ , which can be easily described in similar way in terms of the same model space H. An argument quite similar to that of [16] and based on identities established in [25] leads to the following Theorem. Theorem 2.9. Let the operator A be as above. Then ˆ−(+) , respectively, one has: (i) For all Im λ < 0 (Im λ > 0) and (˜ g, g) ∈ N 1 g˜ g˜ . (2.7) = PK (A − λ)−1 PK k−λ g g Conversely, the property (2.7) for Im λ < 0 (Im λ > 0) guarantees that the ˆ−(+) . vector (˜ g , g) belongs to the set N ˜± ≡ PK N ˆ± holds: (ii) The following non-model description of linear sets N ˜+(−) = u ∈ H : X+(−) αΓ2 (A − λ)−1 u ∈ H 2 (E) . N (2.8) +(−) Here X+(−) αΓ2 (A − λ)−1 u is2 treated as an analytic vector function of λ ∈ C+(−) taking values in the auxiliary Hilbert space E.

Here and throughout the rest of the paper we use the same notation for the objects of the original Hilbert space H and their model representations. We hope that this convention will not confuse the reader. We remark that it can be verified [16] that the projections X± can be dropped altogether in the definition (2.8). The existence of this description gives ground to 2 That is, analytic continuations of the vector X −1 u from the domain of analyticity +(−) αΓ2 (A−λ) of the resolvent to the half-plane C+(−)

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˜+ , N ˜− and N ˜e “smooth” (in the calling the vectors belonging to the linear sets N corresponding half-planes) vectors of the operator A (see also [16]). The proof of Theorem 2.9 will appear in our forthcoming publication.

3. Similarity problem for absolutely continuous extensions of symmetric operators Definition 3.1. We call a non-self-adjoint operator A acting in Hilbert space H an operator with absolutely continuous spectrum if its absolutely continuous subspace Ne (A) coincides with H. Remark 3.2. It has to be noted that the requirement Ne (A) = H doesn’t actually guarantee that the spectrum of the operator A is purely absolutely continuous. Due to the possibility that the absolutely continuous and singular spectral subspaces may intersect, one might face the following situation: Ne (A) = H; Ni (A) = {0} and Ni (A) ⊂ Ne (A). See [31] for one rather transparent example of this case. Nevertheless, due to the fact that Ni (A∗ ) ≡ H ⊖ Ne (A), one easily sees that in the situation of Ne (A) = H the singular spectral subspace of the adjoint operator A∗ is trivial. Let A0 be as above a closed densely defined linear symmetric operator with equal (finite or infinite) deficiency indices n+ (A0 ) = n− (A0 ) ≤ ∞. Let (H, Γ1 , Γ2 ) be a boundary triple for A∗0 . Consider operators A˜B ≡ A, being almost solvable non-self-adjoint extensions of operator A0 . Let the spectrum σ(A) of the operator A be a subset of the real axis.3 Furthermore, without loss of generality we restrict ourselves to consideration of completely non-self-adjoint extensions A only [25, 16]. As above we denote by Θ(λ), S(λ) the characteristic operator-functions of operators A, A+ ≡ A˜B+ , respectively. Then the following theorem holds: Theorem 3.3. Provided that the operator A has absolutely continuous spectrum, the following assertions are equivalent: (i) Operator A is similar to a self-adjoint one (ii) For all u ∈ H the following estimates hold: ((Θ(k − i0)JΘ∗ (k − i0) − J)X+ αΓ2 (A∗+ − k − i0)−1 u, 3 The

X+ αΓ2 (A∗+ − k − i0)−1 u)dk ≤ Cu2

((J − Θ∗ (k + i0)JΘ(k + i0))X− αΓ2 (A∗+ − k − i0)−1 u, X− αΓ2 (A∗+ − k − i0)−1 u)dk ≤ Cu2 .

latter is necessary for operator A to be similar to self-adjoint one.


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Proof. It is proved in [17, 5], that a non-self-adjoint operator A acting in Hilbert space H, the spectrum σ(A) of which is a subset of real axis, is similar to a selfadjoint operator if and only if the following estimates hold: ∞ (A − k − iε)−1 u2 dk ≤ Cu2 sup ε ε>0 −∞ (3.1) ∞ sup ε (A∗ − k − iε)−1 u2 dk ≤ Cu2 ε>0

−∞

for every u ∈ H. Our first goal is to rewrite these estimates in terms of the functional model for non-self-adjoint extensions of symmetric operators described in Section 2. We shall see that provided that the spectrum of the operator in question is absolutely continuous it is possible to compute the limits of the left-hand sides in the resulting estimates when ε → 0 precisely, thus making it possible to replace (3.1) with much more simple conditions, formulated in terms of objects defined on the real line only. This passage from the uniform integral estimates in the half-plane to the integral estimates on the boundary values simplifies matters simplifies matters in the case of one-dimensional Friedrichs model (see [19, 11], cf. [17]). As shown in [11], our approach reduces the similarity problem to the question of boundedness of certain singular integral operators. The latter could be examined by standard methods of analysis yielding necessary and sufficient conditions for the similarity of the operator of the Friedrichs model to a self-adjoint one. It is also worth mentioning that in the process of this limit procedure we reduce the consideration to analytic functions taking values in Hilbert spaces of (potentially) lower dimensions. The linear manifold N ⊂ H has been introduced in the previous section as g + S ∗ g) + X+ (S˜ g + g) = 0} (cf. (2.5)) Since A is an follows: N ≡ {(˜ g , g) : X− (˜ operator with absolutely continuous spectrum, PK N is a dense set in the Hilbert space H. Moreover, for every (˜ g, g) ∈ N and for all λ0 ∈ C+ one has: 1 g˜ g˜ (A − λ0 )−1 PK = PK k − λ0 g g ⎛ ⎞ g˜ + S ∗ g P+ ⎜ g˜ 1 k − λ0 ⎟ ⎟ −⎜ = ⎝ S˜ g+g ⎠ k − λ0 g P− k − λ0

1 g + S ∗ g)(λ0 ) 1 g˜ P+ (˜ = . (3.2) + PK k − λ0 k − λ0 −P+ (S˜ g + g)(λ0 ) g Note that the first estimate in (3.1) in the model terms can be rewritten as follows: ; ;2 ;2 ; ; ; ; g˜ ; ; dk ≤ C ;PK g˜ ; , sup ε ; (A − k − iε)−1 PK ; ; ; g g ; ε>0

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Since the first summand on the right-hand side of (3.2) obviously satisfies it (due to the self-adjointness of the mapping (˜ g, g) → k(˜ g , g)), is sufficient to consider the second term in (3.2) only. Taking into account that S(λ) is an analytic contractive operator-valued function in C+ , we obtain [9, 30] (λ0 ≡ x + iε):

;2 ; ; 1 g + S ∗ g)(λ0 ) ; P+ (˜ ; dℜλ0 Im λ0 ; ; k − λ0 −P+ (S˜ g + g)(λ0 ) ;H 1 = ε dx dk (x − k)2 + ε2

B

A P+ (˜ g + S ∗ g)(x + iε) g + S ∗ g)(x + iε) P+ (˜ I S∗ , × S I −P+ (S˜ −P+ (S˜ g + g)(x + iε) g + g)(x + iε)

A g + S ∗ g)(x + iε) P+ (˜ I S ∗ (x + iε) , = π dx S(x + iε) I −P+ (S˜ g + g)(x + iε)

B P+ (˜ g + S ∗ g)(x + iε) −P+ (S˜ g + g)(x + iε) ∗ g + S g)(x + iε)2 + P+ (S˜ g + g)(x + iε)2 = π dx(P+ (˜ − 2ℜ(S(x + iε)P+ (S˜ g + g)(x + iε), P+ (˜ g + S ∗ g)(x + iε)),

the latter expression having a limit when ε → 0 equal [9] to ;

;2 ; g˜ + S ∗ g ; ; . π; P + ; −(S˜ g + g) ;H

Thus we have proved, that the first inequality in (3.1) is equivalent to the following one: ; ;

;2 ;2 ; ; ; g˜ + S ∗ g ; ; ≤ C ;PK g˜ ; for all g˜ ∈ N. ;P+ (3.3) ; ; ; ; −(S˜ g + g) H g H g

Analogous computations can be carried out for the second estimate in (3.1). The only difference with the case considered above will arise due to the fact, that smooth vectors for operator A are no longer smooth when the operator A∗ is concerned; hence we have to use the general formula for the action of (A∗ − λ)−1 in the model representation (2.4) instead of (2.7). The following assertion holds: the second inequality in (3.1) is equivalent to ; ;

;2 ;2 ; ; ; ; P+ (˜ g + S ∗ g) − c(k) ; ; ≤ C ;PK g˜ ; , (3.4) ; −P+ (S˜ ; g + g) + S(k)c(k) ;H g ;H where

T (λ) ≡ [X+ + X− S(λ)]−1 ,

g + S ∗ g)(λ) + P+ (S˜ g + g)(λ)), c(λ) ≡ T (λ)(P+ (˜

and c(k) ≡ c(k + i0), the boundary values of c(λ) almost everywhere on the real line existing due to an argument from [16] for all (˜ g , g) ∈ N .


277

To complete the proof we now need to rewrite estimates (3.3),(3.4) in terms of objects defined in the “original” Hilbert space H. This can be carried out using another representation of the functional model for the operator A+ , unitarily equivalent to the one used above [25, 16]. Namely, consider the model space H′ defined as follows: H′ ≡ D− ⊕ H ⊕ D+ , where D± ≡ L2 (R± ; E), E = clos Ran(α). The explicit formula describing the action of the operator A+ in H′ was obtained in [25]. The isometric mapping Φ of H′ onto H has the following form4 [25]: 1

v− (k) + vˆ+ (k) g˜ + S ∗ g = −(2π)− 2 αΓ2 (A+ − k + i0)−1 u + S ∗ (k)ˆ 1

v+ (k), S˜ g + g = −(2π)− 2 αΓ2 (A∗+ − k − i0)−1 u + vˆ− (k) + S(k)ˆ

(3.5)

where (v− ; u; v+ ) ∈ H′ ,

1 eikξ v± (ξ)dξ vˆ± (k) = √ 2π is the Fourier transform of the function v± (ξ) extended by zero to the complementary half-line. The boundary values in the strong sense on the real line (∗) αΓ2 (A+ − k ± i0)−1 u exist and belong to H2∓ (E) [25]. The mappings (3.5) together with the absolute continuity of the spectrum of operator A make it possible to complete the proof. We omit here the corresponding purely technical calculations based on identities established in [25] due to the fact that they are similar to those carried out in proving results of [10]. Corollary 3.4. The following estimates are sufficient for the operator A with absolutely continuous spectrum to be similar to a self-adjoint operator: X+ (Θ(k − i0)JΘ∗ (k − i0) − J)X+ ≤ C,

X− (J − Θ∗ (k + i0)JΘ(k + i0))X− ≤ C

for almost all k ∈ R.

This assertion immediately follows from the estimates (see [16, 25]): √ αΓ2 (A+ − λI)−1 uH − (E) ≤ 2πu 2 √ αΓ2 (A∗+ − λI)−1 uH + (E) ≤ 2πu 2

for all u ∈ H. Based on the results, formulated in Theorem 3.3 and on the following identity [25] αΓ2 (A∗+ − λI)−1 = Θ+ (λ)αΓ2 (A − λI)−1 , λ ∈ C+ , where as above Θ+ (λ) ≡ X− +S(λ)X+ , one can rewrite conditions of Theorem 3.3 thus obtaining the following 4 Under

+(−)

this mapping the subspace D+(−) is mapped onto H2 orem [9]

(E) by the Paley-Wiener the-

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Theorem 3.5. Provided that the operator A has absolutely continuous spectrum, the following assertions are equivalent: (i) Operator A is similar to a self-adjoint one (ii) For all u ∈ H the following estimates hold: ((I − S ∗ (k)S(k))X+ αΓ2 (A − k − i0)−1 u, X+ αΓ2 (A − k − i0)−1 u)dk ≤ Cu2 ((I − S ∗ (k)S(k))X− αΓ2 (A∗ − k − i0)−1 u, X− αΓ2 (A∗ − k − i0)−1 u)dk ≤ Cu2 Since the operator-function S(λ) is contractive in the upper half-plane, this immediately yields the following Corollary 3.6. The following estimates are sufficient for the operator A with absolutely continuous spectrum to be similar to a self-adjoint operator: ; ; ;X+ αΓ2 (A − k − i0)−1 u;2 dk ≤ Cu2 ; ; ;X− αΓ2 (A∗ − k − i0)−1 u;2 dk ≤ Cu2

4. Similarity problem for restrictions to invariant spectral subspaces The spectral projection Pδ to the portion δ of the absolutely continuous spectrum was constructed in model terms in [15]. Namely, in the setting of completely nonself-adjoint almost solvable extensions of symmetric operators with equal deficiency indices the following Theorem holds (the proof for additive perturbations of selfadjoint operators can be found in [15]; it is easily extended to the setting of the present paper using results of [25]): Theorem 4.1. Suppose that A is a completely non-self-adjoint operator with absolutely continuous spectrum. For any Borel set δ ⊂ R put g˜ g˜ Pδ PK , (4.1) = PK Xδ g g g˜ where g ∈ N and Xδ is the operator of componentwise multiplication by the characteristic function of the set δ. For the operator Pδ defined by (4.1) on the set ˜e the following assertions hold: of smooth vectors N ˜ ˜ (i) Pδ Ne ⊂ Ne ; (ii) (A − λ0 )−1 Pδ = Pδ (A − λ0 )−1 , Im λ0 = 0; (iii) Pδ Pδ′ = Pδ∩δ′ , δ, δ ′ ⊂ R; ˜e (in fact, as 1 − Xδ → 0 in L∞ (R)); ∞), u ∈ N (iv) Pδ u −→ u as δ → (−∞,

1 −1 ˜e . − (A − k + iε)−1 ]udk, u ∈ N (v) Pδ u = limε→+0 2πi δ [(A − k − iε)


279

We remark that the assertion (v) above establishes the connection between the definition of a spectral projection in terms of the functional model with the usual approach to the definition of spectral projections based on the Riesz integral for the resolvent, and thus the term “spectral projection” is justified. Based on this result, in the present section we obtain conditions, necessary and sufficient for the restrictions of the non-self-adjoint operator A of the class considered in this paper to its spectral subspaces (i.e., to subspaces of the form clos Pδ Ne ) to be similar to self-adjoint operators. We again assume that the operator A is an operator with absolutely continuous spectrum. In essence, Theorem 4.1 allows to repeat the argument of the previous section in relation to any spectral subspace of the form Xδ gg˜ , where gg˜ ∈ N . The characteristic function Xδ of the Borel set δ will then manifest itself in the estimates (3.1) rewritten in model terms, see (3.3), (3.4) by changing the interval over which the integral is being taken. Namely, the integral over the real line in the definition of the H-norm will in all cases be replaced by the integral over the Borel set δ, which together with some obvious modifications immediately yields desired result. For the sake of completeness, we provide here the corresponding results, directly analogous to the results of Section 3. Theorem 4.2. Provided that the operator A has absolutely continuous spectrum, the following assertions are equivalent: (i) The restriction of the operator A to its spectral invariant subspace, corresponding to a Borel set δ ⊂ R, is similar to a self-adjoint operator acting in clos Pδ Ne . (ii) For all u ∈ clos Pδ Ne the following estimates hold: ((Θ(k − i0)JΘ∗ (k − i0) − J)X+ αΓ2 (A∗+ − k − i0)−1 u, δ

δ

X+ αΓ2 (A∗+ − k − i0)−1 u)dk ≤ Cu2

((J − Θ∗ (k + i0)JΘ(k + i0))X− αΓ2 (A∗+ − k − i0)−1 u, X− αΓ2 (A∗+ − k − i0)−1 u)dk ≤ Cu2 .

Theorem 4.3. Provided that operator A has absolutely continuous spectrum, the following assertions are equivalent: (i) The restriction of the operator A to its spectral invariant subspace, corresponding to a Borel set δ ⊂ R, is similar to a self-adjoint operator acting in clos Pδ Ne . (ii) For all u ∈ clos Pδ Ne the following estimates hold: ((I − S ∗ (k)S(k))X+ αΓ2 (A − k − i0)−1 u, X+ αΓ2 (A − k − i0)−1 u)dk ≤ Cu2 δ ((I − S ∗ (k)S(k))X− αΓ2 (A∗ − k − i0)−1 u, X− αΓ2 (A∗ − k − i0)−1 u)dk ≤ Cu2 δ

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Corollary 4.4. The following estimates are sufficient for the restriction of the operator A with absolutely continuous spectrum to its spectral invariant subspace, corresponding to a Borel set δ ⊂ R, to be similar to a self-adjoint operator: X+ (Θ(k − i0)JΘ∗ (k − i0) − J)X+ ≤ C,

X− (J − Θ∗ (k + i0)JΘ(k + i0))X− ≤ C

for almost all k ∈ δ.

Corollary 4.5. The following estimates are sufficient for the restriction of the operator A with absolutely continuous spectrum to its spectral invariant subspace, corresponding to a Borel set δ ⊂ R, to be similar to a self-adjoint operator: ; ; ;X+ αΓ2 (A − k − i0)−1 u;2 dk ≤ Cu2 δ ; ; ;X− αΓ2 (A∗ − k − i0)−1 u;2 dk ≤ Cu2 δ

for all u ∈ clos Pδ Ne .

5. Application to non-self-adjoint extensions of the Laplace operator In this section we demonstrate how the results obtained above apply to the study of the similarity problem for operators that arise when considering Schr¨ odinger equations with potentials of zero radius in quantum mechanics. Namely, following [25], consider a finite set of points {xs }ns=1 , n < ∞ in R3 and symmetric operator A0 being the closure of the Laplace operator −∆ defined on the manifold C0∞ (R3 \ ∪xs ). It is well known [2, 6, 1] that D(A0 ) = {u ∈ W22 (R3 ) : u(xs ) = 0, s = 1, 2, . . . , n}.

The deficiency indices of the operator A0 are equal to (n, n). The following result from [22, 1] makes it possible to describe the domain D(A∗0 ) and to construct a boundary triple for the operator considered: D(A∗0 ) = {u ∈ L2 (R3 ) ∩ W22 (R3 \ {xs }ns=1 )

with the following asymptotic behaviour when x is close to xs : us− + us0 + O(|x − xs |1/2 ), s = 1, 2, . . . , n} u(x) ∼ |x − xs | Moreover, for all u, v ∈ D(A∗0 ) the following equality holds: (A∗0 u, v)H − (u, A∗0 v)H =

n s=1

s − us v s ). (us0 v− − 0

It is shown in [25], that H ≡ Cn and the mappings Γ1 , Γ2 defined as Γ1 u ≡ (u10 , u20 , . . . , un0 ), Γ2 u ≡ (u1− , u2− , . . . , un− ), u ∈ D(A∗0 )


281

can be chosen as a boundary triple for the operator A∗0 . In this case the Weyl function (2.1) M (λ) of A0 is a (n×n) matrix-function with the matrix elements Msj / ik, s=j , s, j = 1, 2, . . . , (5.1) Msj (λ) = eik|xs −xj | , s = j |xs −xj | √ where k = λ, Im k > 0. In [25] it is proved that the characteristic function of the almost solvable extension AB corresponding to the n by n matrix B can be calculated as follows: ΘAB (λ) = I + iJα(B ∗ − M (λ))−1 α

ΘA∗B (λ) = I − iJα(B − M (λ))−1 α, where as before α = 2| Im B| and J = sign Im B. Thus the explicit formula for the Weyl function M (λ) given above permits one to immediately verify both conditions of Corollary 3.4. In the case considered this amounts to nothing more than verifying uniform boundedness of two n by n matrix-functions defined on the real line. Another way to apply the results of Section 3 is the following reformulation of Corollary 3.6. It can be obtained based on the formula from [25] establishing the correspondence between the resolvent of any almost solvable extension AB and the resolvent of the (self-adjoint) Friedrichs extension A∞ in terms of the matrix B and the Weyl function M (λ). Theorem 5.1. Let the operator AB be as defined in the present section. Let further its spectrum be real. Then if for all u ∈ H ; ; ;α(B − M (k + i0))−1 Γ1 (A∞ − k − i0)−1 u;2 dk ≤ Cu2 ; ; ;α(B ∗ − M (k + i0))−1 Γ1 (A∞ − k − i0)−1 u;2 dk ≤ Cu2 ,

where A∞ is the Friedrichs extension of the operator A0 , i.e., the self-adjoint operator −∆ defined on the domain D(A∞ ) = W22 (R3 ), then the operator A is similar to a self-adjoint operator. Further concise results on the similarity problem for this and other applications, together with the proof of the last theorem, will be a subject of our forthcoming publication. Acknowledgements The author is grateful to Prof. S.N. Naboko and Dr. M.M. Faddeev for their constant attention to his work and for valuable discussions of the subject. The author is also grateful to the referee for a number of extremely helpful remarks that hopefully have helped to improve the quality and overall readability of the paper.

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References [1] Albeverio S., Kurasov P., Singular perturbations of differential operators, Cambridge Univ. Press, 2000. [2] Berezin F.A., Faddeev L.D., A remark on Schr¨ odinger’s equation with a singular potential, Soviet Math. Dokl. 2 (1961). [3] Brodskij M.S., Triangular and Jordan representations of linear operators, English transl. in Amer. Math. Soc., Providence, R.I., 1971. [4] Bruk V.M., A certain class of boundary value problems with a spectral parameter in the boundary condition. Mat. Sb. (N.S.) vol. 100 (142), no. 2 (1976), pp. 210–216. [5] Van Casteren J., Boundedness properties of resolvents and semigroups of operators, Acta Sci. Math. Szeged., vol. 48 (1980), N 1–2. [6] Demkov Y.N., Ostrovskii V.N., The use of zero-range potentials in atomic physics, Nauka, Moskva, 1975 (in Russian). [7] Derkach V.A. and Malamud M.M., Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. vol. 95 (1991), pp. 1–95. [8] Gorbachuk V.I. and Gorbachuk M.L., Boundary value problems for operator differential equations. Translated and revised from the 1984 Russian original. Mathematics and its Applications (Soviet Series), 48. Kluwer Academic Publishers Group, Dordrecht, 1991. [9] Hoffman K., Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. [10] Kiselev A.V., Faddeev M.M., The similarity problem for non-self-adjoint operators with absolutely continuous spectrum, Func. Anal. App. vol. 34 (2000), no. 2, pp. 140–142. [11] Kiselev A.V., Some Spectral Properties of the Non-Self-Adjoint Friedrichs Model Operator, Math. Proc. of Royal Irish Academy, vol. 105A (2), 2005. [12] Koˇcube˘ıA.N., On extension of symmetric operators and symmetric binary relations Math. Notes vol. 17 (1975), pp. 41–48 [13] Lax P.D., Phillips R.S., Scattering theory, Pure and Applied Mathematics, 26. Academic Press, Inc., Boston, MA, 1989. [14] Naboko S.N., Absolutely continuous spectrum of a nondissipative operator and functional model. I., Zapiski Nauchnykh Seminarov LOMI, Vol. 65 (1976), pp. 90–102 (Russian); English translation in J. Sov. Math. [15] Naboko S.N., Absolutely continuous spectrum of a nondissipative operator and functional model. II., Zapiski Nauchnykh Seminarov LOMI, Vol. 73 (1977), pp. 118–135 (Russian); English translation in J. Sov. Math. [16] Naboko S.N., Functional model of perturbation theory and its applications to scattering theory, Proc. Steklov Inst. Math. (1981), Issue 2., pp. 85–116. [17] Naboko S.N., The conditions for similarity to unitary and self-adjoint operators, Functional Analysis and its applications, vol. 18 (1984), pp. 16–27. [18] Nikolski N.K., Operators, Functions and Systems: An Easy Reading, vol. I, II, AMS, 2002.


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[19] Nikolski N., Treil S., Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator, Journal d’Analyse Mathématique 87 (2002). [20] Pavlov B.S., Conditions for separation of the spectral components of a dissipative operator, Math. USSR-Izv., vol. 39 (1975), pp. 123–148. [21] Pavlov B.S., Dilation theory and spectral analysis of non-self-adjoint differential operators, AMS Transl., vol. 115 (1980), no. 2. [22] Pavlov B.S., A model of zero-radius potential with internal structure. (Russian) Theoret. Math. Phys. vol. 59, no. 3 (1984), pp. 544–550. [23] Petrov A.M., Spectral projectors of dissipative operators, J. Soviet Math., vol. 57 (1991), no. 5, pp. 3440–3443. [24] Ryzhov V.A., Absolutely continuous and singular subspaces of a non-self-adjoint operator, J. Math. Sci., vol. 87 (1997), no. 5, pp. 3886–3911. [25] Ryzhov V., Functional model of a class of non-self-adjoint extensions of symmetric operators, Operator Theory: Adv. Appl. vol. 174 (2007). [26] Ryzhov V.A., Functional model of nondissipative operator and the scattering problems, Ph.D. thesis, St. Petersburg State University, 1996 (in Russian). [27] Sahnovi˘c L.A., Nonunitary operators with absolutely continuous spectrum, Izv. Akad. Nauk SSSR Ser. Mat. vol. 33 (1969), pp. 52–64 (Russian); English translation in Math. USSR-Izv. ˇ [28] Straus A.V., Characteristic functions of linear operators. Izv. Akad. Nauk. SSSR, Ser. Mat., vol. 24, 1 (1960), pp. 43–74; English transl. in AMS Transl. vol. (2) 40 (1964), pp. 1–37. ˇ [29] Straus A.V., On extensions and characteristic function of a symmetric operator. Izv. Akad. Nauk SSSR Ser. Mat. vol. 32 (1968), pp. 186–207; English transl. in Math. USSR - Izvestija, vol. 2 (1968), pp. 181–204. [30] Sz.-Nagy B. and Foia¸s C., Analyse harmonique des op´ erateurs de l’espace de Hilbert, Masson, Paris and Akad. Kiadó, Budapest, 1967. [31] Veselov V.F., Naboko S.N., The determinant of the characteristic function and the singular spectrum of a non-self-adjoint operator, Mat. Sb. vol. 129 (171) (1986), no. 1, pp. 20–39 (Russian); English translation in Math. USSR-Sb. Alexander V. Kiselev Dept. of Pure Mathematics and Mathematical Physics St.Petersburg State University 1 Ulianovskaya st. 198504 Peterhoff, St.Petersburg, Russia e-mail: [email protected]


Inverse Scattering Problem for a Special Class of Canonical Systems and Non-linear Fourier Integral. Part I: Asymptotics of Eigenfunctions S. Kupin, F. Peherstorfer, A. Volberg and P. Yuditskii Abstract. An original approach to the inverse scattering for Jacobi matrices was recently suggested in [20]. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the inverse scattering problem for a class of canonical systems that generalizes the case of SturmLiouville (Schr¨ odinger) operator. Mathematics Subject Classification (2000). Primary: 34L40,42C05; Secondary: 34L25, 81U40. Keywords. Inverse scattering problem, Schr¨ odinger operator, canonical system, model space, reproducing kernels, asymptotical behavior of the solutions.

1. Faddeev-Marchenko space in Szeg˝ o/Blaschke setting One of the important aspects of the spectral theory of differential operators is the scattering theory [16, 17] and, in particular, the inverse scattering [14]. An original approach to the inverse scattering was recently suggested in [20]. The paper focused on classical Jacobi matrices and connections between the scattering and properties of a special Hilbert transform. Partially supported by NSF grant DMS-0501067 the Austrian Founds FWF, project number: P16390–N04 and Marie Curie International Fellowship within the 6th European Community Framework Programme, Contract MIF1-CT-2005-006966.

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S. Kupin, F. Peherstorfer, A. Volberg and P. Yuditskii

In this paper, we carry out the plan of [20] in the continuous situation. Compared with [20], a completely new feature is that the scattering data incorporate the pure point spectrum with infinitely many mass points. Of course, this is a natural and important step in developing the theory. The discussion leads us to the solution of the inverse scattering problem for a class of canonical systems that include the Sturm-Liouville (Schr¨ odinger) equations. At present, though, we are unable to characterize the scattering data corresponding to the last important special case. The article is quite close to the circle of ideas of [1]–[4], treating the inverse monodromy problem and the inverse spectral problem for a class of canonical systems, and [19], working with a model where the pure point spectrum is not permitted. We also mention an extremely interesting recent paper [8]. The present part of the work is mainly devoted to the asymptotic behavior of certain reproducing kernels (the generalized eigenfunctions). It is organized as follows. Section 1 contains definitions, some general facts and formulations of results on asymptotics. The asymptotic properties of reproducing kernels from certain model spaces are studied in Section 2. Special operator nodes arising from our construction are discussed in Sections 3 and 4. One of the nodes generates a canonical system we are interested in. Its properties and connections to the de Branges spaces of entire functions [5] are also in Section 4. The Sturm-Liouville (Schr¨ odinger) equations are considered in Section 5. An example is given in the first appendix (Section 6). The second appendix (Section 7) relates the whole construction to the matrix A2 Hunt-Muckenhoupt-Wheeden condition. We define the L2 -norm on the real axis as 1 |f (λ)|2 dλ, (1.1) ||f ||2 = 2π R so that the reproducing kernel of the H 2 subspace is of the form k(λ, λ0 ) = λ−iλ¯ 0 . The section “Inverse scattering problem on the real axis” in [14, Chap. 3, Sect. 5] begins with a Sturm-Liouville operator −y ′′ + q(x)y = λ2 y,

x ∈ R,

with the potential q satisfying the a priori condition (1 + |x|)|q(x)| dx < ∞.

(1.2)

(1.3)

R

To such an operator one associates the so-called scattering data {s+ , ν+ },

(1.4)

where s+ is a contractive function on the real axis, |s+ (λ)| ≤ 1, λ ∈ R, possessing certain properties and ν+ is a discrete measure, in fact, supported on a finite number of points Λ = {λk } of the imaginary axis, λik > 0. We proceed in the opposite direction starting from the scattering data {s+ , ν+ } and going to the potential q. The key point of the construction is that

Inverse Scattering Problem and Canonical Systems

287

we assume that the scattering data (1.4) satisfy only very natural (and minimal) conditions from the point of view of the function theory. Namely, we suppose that: ¯ satisfies the Szeg˝o – a symmetric on the real axis function s+ , s+ (λ) = s+ (−λ), condition log(1 − |s+ (λ)|2 ) dλ > −∞, (1.5) 1 + λ2 R – the support Λ of a discrete measure ν+ = k ν+ (λk )δλk satisfies the Blaschke condition λk i < ∞, < ∞. (1.6) i λk (λk /i)>1

(λk /i)≤1

Let us point out that we did not even assume that the measure ν+ is finite. Our plan is to show that already in this case one can associate a certain differential operator of the second order to the given spectral data and then one can prove several specification theorems. Definition 1.1. An element f of the space L2{s+ ,ν+ } is a function on R ∪ Λ such that ||f ||2{s+ ,ν+ } = |f (λk )|2 ν+ (λk ) λk ∈Λ

+ is finite.

1 4π

+ f (λ) R

¯ −f (−λ)

,#

1 s+ (λ)

s+ (λ) 1

$#

$ f (λ) dλ ¯ −f (−λ)

(1.7)

Using (1.5) and (1.6) we define the outer in the upper half-plane function se as |se (λ)|2 + |s+ (λ)|2 = 1 a.e. on R,

(1.8)

(1.9)

and the Blaschke product

B(λ) =

bλk (λ),

k

where bλk (λ) = put

where

λ − λk λ − λk ¯k if λk /i ≤ 1 and bλk (λ) = − λ − λ ¯ k if λk /i > 1. We also λ−λ $ # s s (λ), λ ∈ R, (1.10) S(λ) = − s s+

s se and s− := − s¯+ . (1.11) B s¯ ¯ = S(λ), and The matrix function S possesses two fundamental properties: S ∗ (−λ) it is unitary-valued. The third one is the meromorphic property of the entry s, which has analytic continuation to the upper half-plane as a function of bounded characteristic with a specific nature, that is, it is a ratio of an outer function and a Blaschke product. s :=

288

S. Kupin, F. Peherstorfer, A. Volberg and P. Yuditskii The measure ν− is defined through s− and ν+ by 2 1 ′ 1 1 (λk ) . = ν+ (λk ) ν− (λk ) s

(1.12)

A reason for these and the following definitions will be clarified in a moment. Set $ # + $ # # +$ f (λ) s 0 sf (λ) (1.13) − (λ) = + ¯ sf −f (−λ) s+ 1 $ # − $ # ¯ −f (−λ) 1 s− (λ) = 0 s f − (λ)

for λ ∈ R and

f − (λk ) = f + (λk ) =

′ 1 (λk )ν+ (λk )f + (λk ), s ′ 1 (λk )ν− (λk )f − (λk ). −i s

−i

(1.14)

for λk ∈ Λ. These formulas define two mappings inverse to each other, (.)+ : L2{s− ,ν− } → L2{s+ ,ν+ } , and (.)− : L2{s+ ,ν+ } → L2{s− ,ν− } . If f + ∈ L2{s+ ,ν+ } , we get f − = (f + )− from the second row of (1.13) and the first row of (1.14). The same applies to f + = (f − )+ , f − ∈ L2{s− ,ν− } with obvious changes. When it does not cause any ambiguity, we will write f ± instead of (f ∓ )± . The corresponding maps will be called ±-mappings. It is not difficult to prove that the ±-mappings are unitary from L2{s+ ,ν+ } to L2{s− ,ν− } and vice versa. For example, due to (1.13) $# $ + ,# ||sf + ||2 + ||sf − ||2 1 f (λ) s+ (λ) 1 ¯ dλ = , f (λ) −f (−λ) ¯ −f (−λ) s+ (λ) 1 4π R 2 (1.15) where f is f + restricted to R and we have the standard L2 (R)-norms in the RHS of the equality. More details on the computation are in [20], Lemma 1.3. It is also important that relations (1.13), (1.14) not only define a duality between these two spaces but, what is more important, a duality between corresponding Hardy subspaces. Actually we give two versions of definitions of Hardy subspaces (in general, 2 is they are not equivalent, see an example in Section 6). By the first one, H{s + ,ν+ } ∞ basically the closure of H with respect to the given norm (1.7). More precisely, N2 let BN1 ,N2 = k=N bλk and B = {BN,∞ }. Saying it differently, a Blaschke 1 +1 ′ product B is in B, if B ′ is a divisor of B such that B/B ′ is a finite Blaschke product. Then (1.16) f = BN,∞ g, g ∈ H ∞ , 2 we denote the closure in L2{s+ ,ν+ } of functions of belongs to L2{s+ ,ν+ } . By H{s + ,ν+ } 2 the form (1.16). Let us point out that every element f of H{s is such that se f + ,ν+ }


289

belongs to the standard H 2 , see (1.15). Therefore, in fact, f (λ) has an analytic continuation from the real axis to the upper half-plane. Moreover, the value f (λ) obtained by this continuation, and f (λk ) which is defined for all λk ∈ Λ, since f is a function from L2{s+ ,ν+ } , still perfectly coincide. The second space also consists of functions from L2{s+ ,ν+ } having an analytic continuation to the upper half-plane. ˆ2 Definition 1.2. A function f ∈ L2 belongs to H if g(λ) := (se f )(λ), {s+ ,ν+ }

{s+ ,ν+ }

λ ∈ R, belongs to the standard H 2 and g (λk ), λk ∈ Λ, f (λk ) = se where in the RHS g and se are defined by their analytic continuation to the upper half-plane. ˆ2 are dual in a certain sense. and H It turns out that spaces H 2 {s+ ,ν+ }

{s+ ,ν+ }

2 Theorem 1.3. Let f + ∈ L2{s+ ,ν+ } ⊖ H{s and let f − ∈ L2{s− ,ν− } be defined by + ,ν+ } − 2 ˆ . In short, we write (1.13), (1.14). Then f ∈ H {s− ,ν− }

2 + 2 ˆ2 (H {s− ,ν− } ) = L{s+ ,ν+ } ⊖ H{s+ ,ν+ } .

Proof. We notice that f

+

∈

L2{s+ ,ν+ }

(1.17)

implies

¯ ∈ L2 , λ ∈ R. (sf − )(λ) = s+ (λ)f + (λ) − f + (−λ)

Since

¯ −B(−λ)h(− ¯ ¯ f + , Bh {s+ ,ν+ } = s+ (λ)f + (λ) − f + (−λ), λ) , h ∈ H 2,

2 it follows from f + ∈ L2{s+ ,ν+ } ⊖ H{s that + ,ν+ }

¯ ∈ H 2. (se f − )(λ) = g(λ) := B(λ)(s+ (λ)f + (λ) − f + (−λ))

Now we calculate the scalar product iB(λ) i f + , {s ,ν } =f + (λk )iB ′ (λk )ν+ (λk ) + se f − , ¯k λ − λk + + λ−λ + ′ =f (λk )iB (λk )ν+ (λk ) + g(λk ) = 0. Therefore, by (1.14) we get f − (λk ) =

g se

(λk ), λk ∈ Λ.

2 ˆ2 and H Both H{s {s+ ,ν+ } are spaces of analytic in the upper half-plane + ,ν+ } functions, so they have reproducing kernels. For µ ∈ C+ , we denote them by k{s+ ,ν+ } (λ, µ) = k{s+ ,ν+ } (., µ), kˆ{s+ ,ν+ } (λ, µ) = kˆ{s+ ,ν+ } (., µ).

Recall also that k(λ, µ) = k(., µ) =

i λ−µ ¯

290


is the reproducing kernel of the standard Hardy space H 2 . The first step is to prove asymptotics for the families {eiλx k{s± e2iλx ,ν± e2iλx } (., λ0 )}x∈R and {eiλx kˆ{s± e2iλx ,ν± e2iλx } (., λ0 )}x∈R with λ0 ∈ C+ . Theorem 1.4. i) The following relations hold true in L2 (R): s eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 ) = seiλx k(., λ0 ) + o(1), s eiλx kˆ{s+ e2iλx , ν+ e2iλx } (., λ0 ) = seiλx k(., λ0 ) + o(1)

(1.18)

as x → +∞. Moreover, ¯ 0 )s eiλx k{s e2iλx , ν e2iλx } (., λ0 ) = eiλx k(., λ0 ) + s− e−iλx k(., −λ0 ) + o(1), s(−λ + + ¯ 0 )s eiλx kˆ{s e2iλx , ν e2iλx } (., λ0 ) = eiλx k(., λ0 ) + s− e−iλx k(., −λ0 ) + o(1) s(−λ + + (1.19) as x → −∞. ii) On Λ, we have: eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 ) = o(1), in

L2ν+ -sense

eiλx kˆ{s+ e2iλx , ν+ e2iλx } (., λ0 ) = o(1)

(1.20)

as x → +∞. Furthermore, for a λk ∈ Λ 1 , ν+ (λk ) 1 ν+ (λk )

lim e−2Im λk x k{s+ e2iλx , ν+ e2iλx } (λk , λk ) =

x→−∞

lim e−2Im λk x kˆ{s+ e2iλx , ν+ e2iλx } (λk , λk ) =

x→−∞

(1.21)

It goes without saying that relations (1.18)–(1.20) correspond to scattering “from +∞ to −∞”; compare these formulas to (0.8), (0.25) from [20]. Scattering in the inverse direction (“from −∞ to +∞”) is described similarly. We give the formulas for the family {e−iλx k{s− e−2iλx ν− e−2iλx } (., λ0 )} only; asymptotics for {e−iλx kˆ{s− e−2iλx ν− e−2iλx } (., λ0 )} are the same. Corollary 1.5. We have s e−iλx k{s− e−2iλx ν− e−2iλx } (., λ0 ) = se−iλx k(., λ0 ) + o(1), −iλx ¯ 0 )s e s(−λ k{s e−2iλx ν e−2iλx } (., λ0 ) = e−iλx k(., λ0 ) −

x → −∞,

−

+ s+ eiλx k(., −λ0 ) + o(1),

2

in L -sense on the real line. As for Λ, e−iλx k{s− e−2iλx ν− e−2iλx } (., λ0 ) = o(1) in L2ν− -sense as x → −∞. As before, for a λk ∈ Λ, lim e2Im λk x k{s− e−2iλx ν− e−2iλx } (λk , λk ) =

x→+∞

1 . ν− (λk )

x → +∞

Inverse Scattering Problem and Canonical Systems We set ˜ λ0 ) = k(.,

291

k(., λ0 ), λ ∈ R, 0, λ ∈ Λ.

Theorem 1.4 follows immediately from the following result. Theorem 1.6. The following relations hold true: ˜ λ0 )||{s ,ν } lim ||eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 ) − eiλx k(., + +

x→+∞

lim

x→+∞

||eiλx kˆ{s+ e2iλx , ν+ e2iλx } (., λ0 )

−e

iλx ˜

k(., λ0 )||{s+ ,ν+ }

= 0,

(1.22)

= 0.

(1.23)

The proof of this theorem is the main purpose of Section 2. Remark 1.7. Relations (1.18)–(1.20) follow easily from Theorem 1.6. ˜ λ0 ) = 0 on Λ, we see Indeed, let us have a look at (1.22). Recalling that k(., ||e

iλx

k{s+ e2iλx , ν+ e2iλx } (., λ0 ) − = +

˜ λ0 )||2 eiλx k(., {s+ ,ν+ }

||eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 ) − eiλx k(., λ0 )||2s+

||eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 )||2ν+ → 0,

as x → +∞, so the first relation in (1.20) is proved. Then we notice that $ # $ $ # # |s|2 0 s¯+ 1 s¯+ s+ 1 . = + s+ 1 1 0 0

This implies that the first summand on the right-hand side of the above equality is || . . . ||2s+ = ||s eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 ) − eiλx k(., λ0 ) ||22 iλx − + ||s e k{s+ e2iλx , ν+ e2iλx } (., λ0 ) s − − e−iλx k(., −λ0 ) + s+ eiλx k(., λ0 ) ||22 . The presence of the first term on the right-hand side shows that we are done with (1.18). To deal with (eiλx k{s+ e2iλx , ν+ e2iλx } (., λ0 ))− s− , we use Lemma 2.2 and its corollary saying lim k{s− e−2iλx (bλ0 b−λ¯

x→+∞

¯

0

¯

−1 } (−λ0 , −λ0 ) )−1 , ν− e−2iλx (bλ0 b−λ ¯ ) 0

(see also Lemma 2.8). Hence, we come to s(λ0 )s e−iλx kˆ{s− e−2iλx (bλ0 b−λ¯ )−1 , ν− e−2iλx (bλ0 b−λ¯ 0

0

=

)−1 }

1 2Im λ0 |s(λ0 )|2

¯ 0 ) + s+ (bλ b ¯ )eiλx k(., λ ¯ 0 ) + o(1) = e−iλx k(., −λ 0 −λ0

as x → +∞. This is the second relation in (1.19) up to changes x → −x, λ0 → ¯ 0 , s− (bλ b ¯ )−1 → s+ and s+ (bλ b ¯ ) → s− . −λ 0 −λ0 0 −λ0 Equalities (1.21) are proved in Corollary 2.5.

292


2. Asymptotics of reproducing kernels 2.1. Definitions and notation In this subsection, we prove several propositions concerning special properties of the reproducing kernels introduced in Section 1. For λ0 ∈ C+ , let kˆ (., λ0 ) ˆ {s ,ν } (., λ0 ) = C {s+ ,ν+ } K + + ˆ k{s+ ,ν+ } (λ0 , λ0 )

k{s ,ν } (., λ0 ) , K{s+ ,ν+ } (., λ0 ) = C + + k{s+ ,ν+ } (λ0 , λ0 )

be their normalized versions. It is also convenient to put K(λ, λ0 ) =

i (2Im λ0 )1/2 k(λ, λ0 ) = ¯0 . ||k(., λ0 )|| λ−λ

2 (x) as the closure of the functions For a fixed x ∈ R we define H{s + ,ν+ }

f (λ) = BN,∞ (λ)g(λ)eiλx ,

g ∈ H ∞ , BN,∞ ∈ B.

2 (0). In the similar way = H{s + ,ν+ } ˆ2 that H {s+ ,ν+ } is related to x = 0.

2 H{s + ,ν+ }

In particular, ˆ2 spaces H {s+ ,ν+ } (x), so It is easy to see that

(2.1)

we define the set of

iλx ˆ 2 ˆ2 H H{s+ e2iλx , ν+ e2iλx } , {s+ ,ν+ } (x) = e

2 2 H{s (x) = eiλx H{s 2iλx , ν e2iλx } , + ,ν+ } +e +

and k{s+ ,ν+ } (λ, λ0 ; x) =

kˆ{s+ ,ν+ } (λ, λ0 ; x) =

¯

eix(λ−λ0 ) k{s+ e2iλx , ν+ e2iλx } (λ, λ0 ), ¯

eix(λ−λ0 ) kˆ{s+ e2iλx , ν+ e2iλx } (λ, λ0 )

are the reproducing kernels of these spaces, respectively. We also have their normalized versions K{s+ ,ν+ } (., λ0 ; x) ˆ {s ,ν } (., λ0 ; x) K +

+

= =

eix(λ−Re λ0 ) K{s+ e2iλx , ν+ e2iλx } (., λ0 ), ˆ {s e2iλx , ν e2iλx } (., λ0 ). eix(λ−Re λ0 ) K +

+

This section is mainly devoted to the proof of asymptotic formulas for both types of kernels as x → +∞. 2.2. Some special properties of the reproducing kernels We comment once again the ±-mappings introduced by formulas (1.13), (1.14). They actually depend on the scattering data {s± , ν± }, although we do not indicate this dependence explicitly in most cases. Lemma 2.1. Let w be a meromorphic function on C such that |w(λ)| = 1 on R ¯ The following diagram is and w(λk ) = 0, ∞, for all λk ∈ Λ. Put w∗ (λ) := w(−λ).

Inverse Scattering Problem and Canonical Systems commutative

293

w

L2{ww∗ s+ ,ww∗ ν+ } D⏐ ⏐⏐ +⏐F–

L2{w−1 w−1 s ,w−1 w−1 ν } − − ∗ ∗

−→ L2{s+ ,ν+ } D⏐ ⏐⏐ +⏐F– w∗−1

(2.2)

L2{s− ,ν− }

−→

Here the horizontal arrows are related to the unitary multiplication operators and the vertical arrows are related to two different ±-duality mappings.

Proof. Note that both w and w∗−1 are well defined on R ∪ Λ. Evidently, wf ∈ L2{s+ ,ν+ } means that f ∈ L2{ww∗ s+ ,ww∗ ν+ } . Since |w(λ)| = 1, λ ∈ R, we have that {w−1 w∗−1 s− , w−1 w∗−1 ν− } are minus-scattering data for {ww∗ s+ , ww∗ ν+ } if {s− , ν− } corresponds to {s+ , ν+ }. In other words, the s-function remains the same for both sets of scattering data. Then we use definitions (1.13), (1.14). Let bλ0 =

λ − λ0 ¯ 0 . Note that (bλ0 )∗ = b−λ¯ 0 . λ−λ

Lemma 2.2. We have

(k{s+ ,ν+ } (λ, λ0 ))− =

¯0 ) kˆ{b−1 b−1¯ s− ,b−1 b−1¯ ν− } (λ, −λ b−1 ¯ 0 (λ) 1 λ 0 −λ 0 λ 0 −λ 0 −λ , ¯ 0 ) 2Im λ0 kˆ −1 −1 ¯ ¯ s(−λ s ,b−1 b−1 ν } (−λ0 , −λ0 ) {b b λ0

− ¯ −λ 0

λ0

(2.3)

− ¯ −λ 0

and, consequently, k{s+ ,ν+ } (λ0 , λ0 )kˆ{b−1 b−1 λ0

−1 −1 ¯ ν− } ¯ s− ,bλ0 b−λ −λ 0 0

¯ 0 , −λ ¯0) = (−λ

1 1 ¯ 0 )s(λ0 ) (2Im λ0 )2 . s(−λ (2.4)

Proof. First we note that the following one-dimensional spaces coincide ¯ 0 )}. −1 −1 (λ, −λ {(k{s ,ν } (λ, λ0 )}− = {b−1¯ kˆ −1 −1 +

−λ0 {bλ0 b−λ ¯ s− ,bλ0 b−λ ¯ ν− }

+

0

0

This follows immediately from Theorem 1.3, but we prefer to give a formal proof. Starting with the orthogonal decomposition 2 2 ⊖ bλ0 H{b {k{s+ ,ν+ } (λ, λ0 )} = H{s + ,ν+ } λ

b ¯ 0 s+ ,bλ0 b−λ ¯ ν+ } 0 −λ 0

we have 2 2 {k{s+ ,ν+ } (λ, λ0 )}− = (H{s )− ⊖ (bλ0 H{b + ,ν+ } λ

b ¯ 0 s+ ,bλ0 b−λ ¯ ν+ } 0 −λ 0

)− ,

or, due to (2.2), 2 2 {k{s+ ,ν+ } (λ, λ0 )}− = (H{s )− ⊖ b−1 ¯ 0 (H{bλ + ,ν+ } −λ

Now we use Theorem 1.3 ˆ2 {k{s+ ,ν+ } (λ, λ0 )}− =(L2{s− ,ν− } ⊖ H {s− ,ν− } )

b ¯ 0 s+ ,bλ0 b−λ ¯ ν+ } 0 −λ 0

2 ⊖b−1 ¯ 0 (L{b−1 b−1 −λ

ˆ 2 −1 −1 ⊖H {b b

ˆ2 =b−1 ¯ (H{b−1 b−1 −λ

ˆ2 ⊖ b−λ¯ 0 H {s− ,ν− } ).

λ0

0

λ0

−1 −1 ¯ ν− } ¯ s− ,bλ0 b−λ −λ 0 0 −1 −1 ¯ ν− } ¯ s− ,bλ0 b−λ −λ 0 0

λ0

)− .

−1 −1 ¯ ν− } ¯ s− ,bλ0 b−λ −λ 0 0

)

294


Thus ˆ (k{s+ ,ν+ } (λ, λ0 ))− = Cb−1 ¯ k{b−1 b−1 −λ 0

λ0

−1 −1 ¯ ν− } ¯ s− ,bλ0 b−λ −λ 0 0

¯ 0 ). (λ, −λ

(2.5)

The essential part of the lemma deals with the constant C. We calculate the scalar product A B iB(λ) k{s+ ,ν+ } (λ, λ0 ), . ¯0 λ−λ {s+ ,ν+ }

2 2 On the one hand, since iB(λ) ¯ 0 belongs to the intersection of L{s+ ,ν+ } with H , we λ−λ can use the reproducing property of k{s+ ,ν+ } : B A ¯0 ) B(λ0 ) iB(λ) B(−λ = = . (2.6) k{s+ ,ν+ } (λ, λ0 ), ¯0 2Im λ 2Im λ0 λ−λ 0 {s+ ,ν+ }

On the other hand we can reduce the given scalar product to the scalar product in the standard H 2 . Since B(λk ) = 0, the ν-component disappears and we get G# $ 9 iB(λ) :H $ # 1 k{s+ ,ν+ } (λ, λ0 ) 1 s¯+ ¯0 λ−λ (λ) ¯ ¯ λ0 ) , iB(−λ) s+ 1 −k{s+ ,ν+ } (−λ, 2 ¯ λ ¯0 λ+ H G iB(λ) = s(λ)(k{s+ ,ν+ } (λ, λ0 ))− , ¯0 . λ+λ Substituting here (2.5) and using s = se /B we come to A ¯ 0 ), b ¯ (λ) C se (λ)kˆ{b−1 b−1¯ s− ,b−1 b−1¯ ν− } (λ, −λ −λ0 λ0

−λ 0

λ0

Since se (λ)kˆ{b−1 b−1¯ λ0

s ,b−1 b−1¯ ν− } −λ 0 − λ 0 −λ 0

−λ 0

i ¯0 λ+λ

B

.

¯ 0 ) belongs to H 2 and b ¯ (λ) (λ, −λ −λ0

i is the reproducing kernel of H 2 , relation (2.6) yields λ + λ0 ¯ 0 )kˆ −1 −1 Cse (−λ {b b λ0

−1 −1 ¯ ν− } ¯ s− ,bλ0 b−λ −λ 0 0

¯ 0 , −λ ¯0) = (−λ

i ¯0 = λ+λ

¯0) B(−λ . 2Im λ0

Thus (2.3) is proved. Comparing the norms of these vectors and taking into account that the −-map is an isometry we get (2.4). As a consequence of the above lemma, we have ˆ2 b−λ¯ 0 (eiλx K{s+ e2iλx , ν+ e2iλx } (., λ0 ))− ∈ H {s− (bλ

b ¯0 ) 0 −λ

−1 , ν (b −1 } ¯ ) − λ0 b−λ 0

Indeed, using diagram (2.2), we get for x ≥ 0

,

x ≥ 0.

iλx (K{s+ e−2iλx , ν+ e−2iλx } (., λ0 ))− (e−iλx K{s+ e−2iλx , ν+ e−2iλx } (., λ0 ))− s− = e s− e2iλx

ˆ {s e2iλx (b b ¯ = C(λ0 )eiλx K − λ 0 −λ

¯

0

)−1 , ν− e2iλx (bλ0 b−λ ¯ )−1 } (., −λ0 ) 0

ˆ2 by (2.3). So, the latter function is in H {s− (bλ

b ¯0 ) 0 −λ

−1 , ν (b ¯ )−1 } − λ0 b−λ 0

.


295

For discrete measures ν± (1.6), let ν±N be their truncations ν±N =

N

k=1

ν± (λk )δλk .

We say few more words about spaces 2 2 ˆ2 ˆ2 ,H H{s {s± ,ν± } and H{s± ,ν±N } , H{s± ,ν±N } . ± ,ν± }

2 ˆ2 Recall that H{s ⊂H {s+ ,ν+ } . + ,ν+ }

Lemma 2.3. i) Let ||s+ ||∞ < 1 (or, what is the same, inf R |se | > 0). Then ˆ2 H2 , =H {s+ ,ν+ }

{s+ ,ν+ }

ˆ {s ,ν } (., λ0 ). and, consequently, K{s+ ,ν+ } (., λ0 ) = K + + ii) We always have ˆ {s ,ν } (λ0 , λ0 ). K{s ,ν } (λ0 , λ0 ) ≤ K +

+

+

+

ˆ {s ,ν } (., λ0 ). The equality above takes place if and only if K{s+ ,ν+ } (., λ0 ) = K + + iii) Obviously, ˆ2 ˆ2 H2 ⊂ H2 , H ⊂H , and

{s+ ,ν+ }

{s+ ,ν+N }

{s+ ,ν+ }

{s+ ,ν+N }

≤ K{s+ ,ν+N } (λ0 , λ0 ), ˆ {s ,ν } (λ0 , λ0 ). ≤ K + +N

K{s+ ,ν+ } (λ0 , λ0 ) ˆ {s ,ν } (λ0 , λ0 ) K + +

As before, the inequalities become equalities if and only if the corresponding reproducing kernels coincide. Proof. To prove i), we only have to show the inverse inclusion. Suppose that f ∈ 2 ∞ 2 ˆ2 H {s+ ,ν+ } . By Definition 1.2, f ∈ L{s+ ,ν+ } and se f ∈ H . Since se , 1/se ∈ H , we 2 2 see f ∈ H and hence f ∈ H{s+ ,ν+ } . The claim about the reproducing kernels is trivial. 2 ˆ2 ⊂H The inequality in ii) of course follows from inclusion H{s {s+ ,ν+ } . + ,ν+ } n ˆ Consider a system {fn }n∈Z+ , fn = b K{s (b b ¯ )n ,ν (b b ¯ )n } (., λ0 ). This is an λ0

+

λ 0 −λ 0

+

λ 0 −λ 0

ˆ2 ˆ2 orthonormal basis in H {s+ ,ν+ } . We have K{s+ ,ν+ } (., λ0 ) ∈ H{s+ ,ν+ } and ||K{s+ ,ν+ } (., λ0 )||{s+ ,ν+ } = 1. So an f n K{s+ ,ν+ } (., λ0 ) = n

ˆ {s ,ν } (λ0 , λ0 ). Obviously, |a0 |2 ≤ 1 and claim ii) is and a0 = K{s+ ,ν+ } (λ0 , λ0 )/K + + proved. Let us have a look at iii). The first inclusion follows from the fact that for f ∈ H 2 (C+ ) ||BN ′ ,∞ f ||{s+ ,ν+N } ≤ ||BN ′ ,∞ f ||{s+ ,ν+ } .

296


ˆ2 The second one follows from Definition 1.2 of H {s+ ,ν+ } . The inequalities for the reproducing kernels are corollaries of these inclusions; to prove them just argue as in ii). Rewriting (2.4), we have K{s+ e−2iλx , ν+ e−2iλx } (λ0 , λ0 )K{s− e2iλx (bλ0 b−λ¯ =

1 . |s(λ0 )|(2Im λ0 )

¯

0

¯

−1 } (−λ0 , −λ0 ) )−1 , ν− e2iλx (bλ0 b−λ ¯ ) 0

under assumptions i) of the above lemma. 2 We denote by P{s+ ,ν+ } the orthogonal projector from L2{s+ ,ν+ } on H{s . + ,ν+ } 2 (x) Furthermore, Px,{s ,ν } and Pˆx,{s ,ν } are orthogonal projectors on H +

+

+

{s+ ,ν+ }

+

ˆ2 and H {s+ ,ν+ } (x), correspondingly.

Lemma 2.4. We have for any f ∈ L2{s+ ,ν+ } : i) ii)

lim Pˆx,{s+ ,ν+ } f = f

(2.7)

lim Pˆx,{s+ ,ν+ } f = 0

(2.8)

lim Px,{s+ ,ν+ } f = f,

x→−∞

lim Px,{s+ ,ν+ } f = 0,

x→+∞

x→−∞ x→+∞

Symbolically, we may say that 2 2 i) lim eiλx H{s 2iλx , ν e2iλx } = L{s ,ν } , + + +e + x→−∞

2 ˆ 2 2iλx lim eiλx H {s+ e , ν+ e2iλx } = L{s+ ,ν+ } ,

x→−∞

2 ˆ 2 2iλx ii) lim eiλx H{s lim eiλx H 2iλx , ν e2iλx } = {0}, {s+ e , ν+ e2iλx } = {0}. +e + x→+∞

x→+∞

Proof. We prove the first equality in (2.8); the argument for the second equality is likewise. Relations in (2.7) drop by duality, since .+ ˆ 2 −2iλx = eiλx H 2 2iλx e−iλx H . L2 2iλx ⊕ −2iλx {s+ ,ν+ }

{s+ e

, ν+ e

}

{s− e

ν− e

}

iλx1 2 2 Obviously, eiλx2 H{s H{s+ e2iλx1 , ν+ e2iλx1 } for x1 ≤ x2 and so 2iλx2 , ν e2iλx2 } ⊂ e +e +

k{s+ ,ν+ } (λ0 , λ0 ; x) = e−2Im λ0 x k{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) is decreasing with respect to x ∈ R. We have to prove that lim e−2Im λ0 x k{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) = 0,

x→+∞

which is trivial since the second factor tends to k(λ0 , λ0 ) by Lemma 2.8. Corollary 2.5. We have lim e−2Im λk x k{s+ e2iλx , ν+ e2iλx } (λk , λk ) =

x→−∞

1 . ν+ (λk )


297

Proof. Let us consider gk = (1/ν+ (λk )) δλk . Recall that ¯

k{s+ ,ν+ } (., λ0 ; x) = eix(λ−λ0 ) k{s+ e2iλx , ν+ e2iλx } (., λ0 ). 2 Hence, we obtain for a f ∈ H{s (x) + ,ν+ }

(f, gk ){s+ ,ν+ } =

f (λk ) ν+ (λk ) = f (λk ). ν+ (λk )

On the other hand, f (λk ) = (f, k{s+ ,ν+ } (., λk ; x)){s+ ,ν+ } = (f, Px,{s+ ,ν+ } gk ){s+ ,ν+ } . By Lemma 2.4, lim ||k{s+ ,ν+ } (., λk ; x)||2{s+ ,ν+ } = ||gk ||2{s+ ,ν+ }

x→−∞

which becomes the claim of the corollary if we write the norms explicitly.

2.3. Proof of Theorem 1.6 Lemma 2.6. We have k{s+ ,ν+ } (., λ0 ) = lim (ε + I + H{s+ ,ν+ } )−1 k(., λ0 ), ε→0+

where H{s+ ,ν+ } is the Hankel operator coming from the metric (1.7) and the limit is understood in L2{s+ ,ν+ } -sense. The argument follows [20], Lemma 1.2, and is omitted. Lemma 2.7. Let ||s+ ||∞ < 1 and ν+ be a measure with a finite support. Then lim

x→+∞

K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) = 1. K(λ0 , λ0 )

Proof. We see that |k{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) − k(λ0 , λ0 )|2

= |(k{s+ e2iλx , ν+ e2iλx } (., λ0 ) − k(., λ0 ), k{s+ e2iλx , ν+ e2iλx } (., λ0 )){s+ e2iλx , ν+ e2iλx } |2 = | (I + H{s+ e2iλx , ν+ e2iλx } ){(I + H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ) − k(., λ0 )}, (I + H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ) |2

= |(H{s+ e2iλx , ν+ e2iλx } k(., λ0 ), (I + H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ))|2 ≤ C |(Hs+ e2iλx k(., λ0 ), (I + H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ))|2 . + |(Hν+ e2iλx k(., λ0 ), (I + H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ))|2

The bound for the first term is easy |...| ≤

1 ¯ ¯ λ0 ))||2 ||k(., λ0 )||2 → 0 ||P+ (s+ e−2iλx k(−λ, 1 − ||s+ ||∞

298


as x → +∞ by the L2 -Fourier theorem. Since F = (I+H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ) √ ∈ H 2 satisfies ||F ||2 ≤ C, we get |F (λk )| ≤ C/ Im λk and |(Hν+ e2iλx k(., λ0 ), (I

+

H{s+ e2iλx , ν+ e2iλx } )−1 k(., λ0 ))|

≤

C

N

k=1

|k(λk , λ0 )| ν+ (λk )e−2Im λk x √ . Im λk

The right-hand side of the inequality goes to 0 as x → +∞.

The following lemma is the main key to the proof of the asymptotics. Lemma 2.8. We have K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) = 1, K(λ0 , λ0 ) ˆ {s e2iλx , ν e2iλx } (λ0 , λ0 ) K + + lim = 1. x→+∞ K(λ0 , λ0 ) lim

(2.9)

x→+∞

(2.10)

Proof. We start with the proof of the first equality. Taking the square root of both sides of (2.4), we see

=

K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) 1 ˆ {s e−2iλx (b 2Im λ0 |s(λ0 )| K − λ

1 ¯

¯

−1 } (−λ0 , −λ0 ) b ¯ 0 )−1 , ν− e−2iλx (bλ0 b−λ ¯ ) 0 −λ 0

≥

1 ˆ {s e−2iλx (b 2Im λ0 |s(λ0 )| K − λ

=

|B(λ0 )| K 2iλx 2iλx (λ0 , λ0 ). |B1,N (λ0 )| {s+ e , ν+N e }

1

¯

¯

(−λ0 , −λ0 ) b ¯ 0 )−1 , ν−N e−2iλx (bλ0 b−λ ¯ )−1 } 0 −λ 0 (2.11)

Then we continue as

K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) ≥

≥

=

|BN,∞ (λ0 )|(k{s+ e2iλx , ν+N e2iλx } (λ0 , λ0 ))1/2

(2.12)

|BN,∞ (λ0 )|((ε + I + H{s+ e2iλx , ν+N e2iλx } )−1 k(., λ0 ))1/2 (λ0 ) 1 √ |BN,∞ (λ0 )|K{ s+ e2iλx , ν+N e2iλx } (λ0 , λ0 ). 1+ε 1+ε 1+ε

Let sN = se /B1,N . We have (2.13) K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) ≤ K{s+ e2iλx , ν+N e2iλx } (λ0 , λ0 ) 1 1 = ¯ 0 , −λ ¯0 ) ˆ {s e−2iλx (b b )−1 , ν e−2iλx (b b )−1 } (−λ 2Im λ0 |sN (λ0 )| K −

≤

¯ λ 0 −λ 0

1 2Im λ0 |sN (λ0 )| K{s− e−2iλx (bλ0 b−λ¯

¯ λ 0 −λ 0

−N

1

¯

0

¯

(−λ0 , −λ0 ) )−1 , ν−N e−2iλx (bλ0 b−λ ¯ )−1 } 0


299

by ii), Lemma 2.3. We set s−,ε = s− /(1 + ε), ν−N,ε = ν−N /(1 + ε); the functions sN,ε , s+,ε are defined by unitarity of the scattering matrix, and ν+N,ε is defined by ν−N,ε through relations (1.12). Notice that the support of ν+N,ε is the same as ˆ are the same the support of ν+N (and equals {λk }k=1,N ). Since K- and K-kernels for pairs {s+,ε , ν+N,ε } and {s−,ε , ν−N,ε } by i), Lemma 2.3, we continue as √ 1+ε 1 (. . .) ≤ ¯ 0 , −λ ¯0) 2Im λ0 |sN (λ0 )| K{s−,ε e−2iλx (bλ0 b−λ¯ )−1 , ν−N,ε e−2iλx (bλ0 b−λ¯ )−1 } (−λ 0 0 √ 1 + ε |sN,ε (λ0 )| K{s+,ε e2iλx , ν+N,ε e2iλx } (λ0 , λ0 ). = |sN (λ0 )| That is,

≤

|BN,∞ (λ0 )| √ K{ s+ e2iλx , ν+N e2iλx } (λ0 , λ0 ) ≤ K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) 1+ε 1+ε 1+ε √ 1 + ε |sN,ε (λ0 )| K{s+,ε e2iλx , ν+N,ε e2iλx } (λ0 , λ0 ). |sN (λ0 )|

The quantities K{ s+

ν 2iλx , +N 1+ε e 1+ε

e2iλx } (λ0 , λ0 )

and K{s+,ε e2iλx , ν+N,ε e2iλx } (λ0 , λ0 ) tend

to K(λ0 , λ0 ) as x → +∞ by Lemma 2.7. Remaining factors in the left- and righthand side parts of the inequality go to 1 with ε → +0, N → +∞. Hence, for any ε′ > 0 we can choose appropriate ε, N to have 1 − ε′

≤ ≤

K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) K(λ0 , λ0 ) K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 ) ≤ 1 + ε′ , lim sup K(λ0 , λ0 ) x→+∞

lim inf

x→+∞

and (2.9) is proved. The proof of (2.10) is almost identical. First of all, to keep the notation we used to, we prove lim

x→+∞

¯ 0 , −λ ¯0) ˆ {s e2iλx , ν e2iλx } (−λ K − − = 1. ¯ 0 , −λ ¯0) K(−λ

(2.14)

¯ 0 → λ0 and instead of (2.10). This is obviously the same thing up to changes −λ ˆ by s− → s+ . The second modification is that we estimate the value of a K-kernel the values of K-kernels (and not vice versa as we have just done to prove (2.9)). So, as in (2.12), we have ¯ 0 , −λ ¯0) ≥ ˆ {s e2iλx , ν e2iλx } (−λ K − − ≥ =

¯ 0 , −λ ¯0) K{s− e2iλx , ν− e2iλx } (−λ 1 ¯ 0 , −λ ¯0) √ K s− 2iλx ν− 2iλx (−λ 1 + ε { 1+ε e , 1+ε e } |BN,∞ (λ0 )| ¯ 0 , −λ ¯ 0 ). √ K{ s− e2iλx , ν−N e2iλx } (−λ 1+ε 1+ε 1+ε

300


The first inequality in the above estimate is ii), Lemma 2.3 and the last one repeats computation (2.11). Similarly to (2.13), we get ¯ 0 , −λ ¯0) ≤ K ¯ 0 , −λ ¯0) ˆ {s e2iλx , ν e2iλx } (−λ ˆ {s e2iλx , ν e2iλx } (−λ K −

= ≤ = =

−

−

−N

1 1 2Im λ0 |sN (λ0 )| K{s+ e−2iλx (bλ0 b−λ¯ )−1 , ν+N e−2iλx (bλ0 b−λ¯ )−1 } (λ0 , λ0 ) 0 0 √ 1+ε 1 2Im λ0 |sN (λ0 )| K{ s+ e−2iλx (bλ b ¯ )−1 , ν+N e−2iλx (bλ b ¯ )−1 } (λ0 , λ0 ) 1+ε 1+ε 0 −λ 0 0 −λ 0 √ 1 + ε |sN,ε (λ0 )| ˆ ¯ 0 , −λ ¯0) K{s−,ε e2iλx , ν−N,ε e2iλx } (−λ |sN (λ0 )| √ 1 + ε |sN,ε (λ0 )| ¯ 0 , −λ ¯0) K{s−,ε e2iλx , ν−N,ε e2iλx } (−λ |sN (λ0 )|

s+ ν+N , 1+ε } as explained after (2.13). Above, the pair {s−,ε , ν−N,ε } comes from { 1+ε Hence, |BN,∞ (λ0 )| ¯ 0 , −λ ¯0) ≤ K ¯ 0 , −λ ¯0) ˆ {s e2iλx , ν e2iλx } (−λ √ K{ s− e2iλx , ν−N e2iλx } (−λ − − 1+ε 1+ε 1+ε √ 1 + ε |sN,ε (λ0 )| ¯ 0 , −λ ¯ 0 ). K{s−,ε e2iλx , ν−N,ε e2iλx } (−λ ≤ |sN (λ0 )| Repeating the argument from the first part of the proof, we see that for every ε′ > 0 ¯ 0 , −λ ¯0 ) ˆ {s e2iλx , ν e2iλx } (−λ K − − 1 − ε′ ≤ lim inf ¯ ¯ x→+∞ K(−λ0 , −λ0 )

≤

lim sup x→+∞

and relation (2.14) is proved.

¯ 0 , −λ ¯0) ˆ {s e2iλx , ν e2iλx } (−λ K − − ≤ 1 + ε′ , ¯ ¯ K(−λ0 , −λ0 )

Proof of Theorem 1.6. At present, the claim of the theorem is an easy consequence of Lemma 2.8. For an arbitrary N , we have ˜ λ0 )||{s ,ν } ||eiλx K{s e2iλx , ν e2iλx } (., λ0 ) − eiλx K(., +

+

+

≤

−

+

BN,∞ eiλx K(., λ0 )||{s+ ,ν+ } ˜ λ0 ) − BN,∞ eiλx K(., λ0 )||{s ||eiλx K(.,

+ ,ν+ }

The claim will then follow if we prove ˜ λ0 ) − BN,∞ eiλx K(., λ0 )||{s lim sup ||eiλx K(., x→+∞

+

||eiλx K{s+ e2iλx , ν+ e2iλx } (., λ0 )

+ ,ν+ }

≤ C1 |1 − BN,∞ (λ0 )|

.

(2.15)

lim sup ||eiλx K{s+ e2iλx , ν+ e2iλx } (., λ0 ) − BN,∞ eiλx K(., λ0 )||{s+ ,ν+ } x→+∞

≤ C2 |1 − BN,∞ (λ0 )|

(2.16)


301

with some constants C1 , C2 . The computation for (2.15) is easy and elementary ˜ λ0 ) − BN,∞ eiλx K(., λ0 )||2 ||eiλx K(., {s+ ,ν+ }

≤ ||eiλx (1 − BN,∞ )K(., λ0 )||2s+ + ||eiλx BN,∞ K(., λ0 )||2ν+

The second term above obviously goes to 0 as x → +∞; for the first one we have ||eiλx (1 − BN,∞ )K(., λ0 )||2s+ ≤ 2||eiλx (1 − BN,∞ )K(., λ0 )||22 ≤ 4|1 − BN,∞ (λ0 )|.

We pass to (2.16) now. Once again, for an arbitrary N , ||eiλx K{s+ e2iλx , ν+ e2iλx } (., λ0 ) − BN,∞ eiλx K(., λ0 )||2{s+ ,ν+ }

≤ ||eiλx K{s+ e2iλx , ν+ e2iλx } (., λ0 )||2{s+ ,ν+ }

− 2Re (K{s+ e2iλx , ν+ e2iλx } (., λ0 ), BN,∞ K(., λ0 )){s+ e2iλx , ν+ e2iλx }

+ ||BN,∞ K(., λ0 )||2{s+ e2iλx , ν+ e2iλx } . By Lemma 2.8, we get for the second term Re (. . .) =

BN,∞ (λ0 )K(λ0 , λ0 ) → BN,∞ (λ0 ) K{s+ e2iλx , ν+ e2iλx } (λ0 , λ0 )

as x → +∞. The third term is (. . .) = ≤

+

||BN,∞ K(., λ0 )||2s+ e2iλx + ||BN,∞ K(., λ0 )||2ν+ e2iλx

¯ 2 ||BN,∞ K(., λ0 )||2 ||K(., λ0 )||22 + ||P+ [s+ e2iλx BN,∞ K(., λ0 )](−λ)||

||BN,∞ K(., λ0 )||2ν+ e2iλx → 1,

since ||K(., λ0 )||22 = 1 and the rest tends to 0 with x → +∞ (for the second term, this is Fourier L2 -theorem). So, summing up lim sup ||eiλx K{s+ e2iλx , ν+ e2iλx } (., λ0 ) − x→+∞

≤

BN,∞ eiλx K(., λ0 )||{s+ ,ν+ } 2Re (1 − BN,∞ (λ0 )),

and (2.16) is proved. The proof of (1.23) is likewise, we just have to use (2.10) instead of (2.9). 2

3. Unitary node, I Consider the multiplication operator by v¯, v =

λ2 − λ20 ¯2 , acting in λ2 − λ 0

+ 2 ˆ2 L2{s+ ,ν+ } = (H {s− ,ν− } ) ⊕ H{s+ ,ν+ } .

(3.1)

Lemma 3.1. The multiplication operator by v¯ acts as a unitary operator from + 2 {kˆ{s (λ, λ0 )} ⊕ H{s (x) (3.2) + ,ν+ } − ,ν− }

to

+ ¯ 0 )} ⊕ H 2 {kˆ{s (λ, −λ {s+ ,ν+ } (x). − ,ν− }

(3.3)

302


Proof. It is obvious that the multiplication by v¯ = {f ∈

ˆ2 H {s− ,ν− }

: f (λ0 ) = 0} =

b−λ ¯ 0 bλ0

acts from

ˆ2 bλ0 H {bλ0 b−λ ¯ s− ,bλ0 b−λ ¯ ν− } 0 0

to ¯ ˆ2 ˆ2 {f ∈ H ¯0 H {s− ,ν− } : f (−λ0 ) = 0} = b−λ {bλ

b ¯ 0 s− ,bλ0 b−λ ¯ ν− } 0 −λ 0

Therefore it acts on their orthogonal complements (3.2), (3.3).

.

We now recall the definition of the characteristic function of a unitary node [6, 7] and its functional model. An extensive discussion of the subject and its application to interpolation problems can be found in [11]–[13]. Let K, E1 , E2 be Hilbert spaces and U be a unitary operator acting from K ⊕ E1 to K ⊕ E2 . We assume that E1 and E2 are finite-dimensional (dim E1 = dim E2 = 1 in this section, and dim E1 = dim E2 = 2 in Section 4). The characteristic function is defined by Θ(ζ) := PE2 U (IK⊕E1 − ζPK U )−1 |E1 .

(3.4)

It is a holomorphic in the unit disk {ζ : |ζ| < 1} contractive-valued operator function. We make a specific assumption that Θ(ζ) has an analytic continuation in the exterior of the unite disk through a certain arc (a, b) ⊂ T by the symmetry principle −1 1 . Θ(ζ) = Θ∗ ¯ ζ For f ∈ K define F (ζ) := PE2 U (I − ζPK U )−1 f.

(3.5)

This E2 -valued holomorphic function belongs to the functional space KΘ with the following properties. • F (ζ) ∈ H 2 (E2 ) and - .it has analytic continuation through the arc (a, b). 2 • F∗ (ζ) := Θ∗ (ζ)F ζ1¯ ∈ H− (E1 ). # $ F∗ (ζ) belongs to the image of the • For almost every ζ ∈ T the vector F # $ ∗ I Θ operator (ζ), and therefore the scalar product Θ I G# $[−1] # $ # $H I Θ∗ F F∗ , ∗ Θ I F F E1 ⊕E2

is well defined and does not depend of the choice of a preimage (the first term in the above scalar product). Moreover, $[−1] # $ # $H G# F I Θ∗ F∗ , ∗ dm < ∞. (3.6) Θ I F F T E1 ⊕E2


303

One can consider the above items as the definition of the space KΘ . The integral in (3.6) represents the square of the norm of F in the space. Note that PK U |K becomes a certain “standard” operator in the model space f → F (ζ)

=⇒

PK U f →

F (ζ) − F (0) , ζ

(3.7)

see (3.5). The following simple identity is a convenient tool in the forthcoming calculation. Lemma 3.2. For a unitary operator U : K ⊕ E1 → K ⊕ E2

U ∗ PE2 U (I − ζPK U )−1 = I + (ζ − U ∗ )PK U (I − ζPK U )−1 .

(3.8)

Proof. Since IK⊕E2 = PK + PE2 and U is unitary we have U ∗ PE2 U = (I − ζPK U ) + (ζ − U ∗ )PK U.

Then we multiply this identity by (I − ζPK U )−1 .

Theorem 3.3. Let e1 , e2 be the normalized vectors in the one-dimensional spaces (3.2) and (3.3), + + ¯ kˆ{s kˆ{s ,ν } (λ, −λ0 ) ,ν } (λ, λ0 ) e1 (λ) = C − − , e2 (λ) = C − − . kˆ{s− ,ν− } (λ0 , λ0 ) kˆ{s− ,ν− } (λ0 , λ0 )

(3.9)

2 is of the form Then the reproducing kernel of H{s + ,ν+ }

k{s+ ,ν+ } (λ, µ) =

(ve2 )(λ)(ve2 )(µ) − e1 (λ)e1 (µ) 1 − v(λ)v(µ)

.

(3.10)

Proof. First, we are going to find the characteristic function of the multiplication operator by v¯ with respect to decompositions (3.2) and (3.3) and the corresponding functional representation of this node. By (3.9) we fixed “bases” in the one-dimensional spaces. So, instead of the operator we get a scalar function θ(ζ): Θ(ζ)e1 := PE2 U (I − ζPK U )−1 e1 = e2 θ(ζ).

(3.11)

We substitute (3.11) in (3.8) v(λ)e2 (λ)θ(ζ) = e1 (λ) + (ζ − v(λ))(PK U (I − ζPK U )−1 e1 )(λ).

(3.12)

+ (λ, λ0 ): it has analytic continuation in Recall an important property of kˆ{s − ,ν− } ¯ 0 (see Lemma 2.2). Therefore all the upper half-plane with the only pole at −λ terms in (3.12) are analytic in λ and we can choose λ satisfying v(λ) = ζ. Then we obtain the characteristic function in terms of the reproducing kernels

θ(v(λ)) =

e1 (λ) . v(λ)e2 (λ)

(3.13)

304

S. Kupin, F. Peherstorfer, A. Volberg and P. Yuditskii 2 Similarly for f ∈ K = H{s we define the scalar function F (ζ) by + ,ν+ }

Using again (3.8) we get

PE2 U (I − ζPK U )−1 f = e2 F (ζ).

(3.14)

v(λ)e2 (λ)F (ζ) = f (λ) + (ζ − v(λ))(PK U (I − ζPK U )−1 f )(λ).

Therefore,

f (λ) . (3.15) v(λ)e2 (λ) Now we are in a position to get (3.10). Indeed, by (3.14) and (3.15) we proved that the vector PK (I − v(µ)U ∗ PK )−1 U ∗ e2 v(µ)e2 (µ) 2 is the reproducing kernel of K = H{s with respect to µ, |v(µ)| < 1. Using the + ,ν+ } Darboux identity F (v(λ)) =

PE2 U (I − ζPK U )−1 PK (I − ζ¯0 U ∗ PK )−1 U ∗ |E2 =

I − Θ(z)Θ∗ (ζ0 ) 1 − ζ ζ¯0

(in this setting this is a simple and pleasant exercise) we obtain k{s+ ,ν+ } (λ, µ) = v(λ)e2 (λ)

I − θ(v(λ))θ(v(µ)) 1 − v(λ)v(µ)

v(µ)e2 (µ)

for |v(λ)| < 1, |v(µ)| < 1. By analyticity and (3.13) we have that relation (3.10) holds for all λ, µ ∈ C+ . Corollary 3.4. The following Wronskian-type identity is satisfied for the reproducing kernels − (se )(µ) (se− )(µ) 1 1 2 = (log v(µ))′ , Im µ > 0. (3.16) e2 (µ) e1 (µ) i

− (., .) instead of (k{s+ ,ν+ } (., .))− . So we multiply Proof. To be brief, we write k{s + ,ν+ } − µ) by bμ (λ) and calculate the resulting function of λ at λ = µ. By k{s+ ,ν+ } (λ, −¯ (2.3) we get 1 − {bμ (λ)k{s (λ, −¯ µ)}λ=μ = . (3.17) + ,ν+ } s(µ)2Im µ Now we make the same calculation using representation (3.10). Since −v(µ) v(λ)e− e− − 2 (λ) 1 (λ) , (λ, −¯ µ ) = k{s + ,ν+ } µ) v(−¯ µ)e2 (−¯ µ) v(λ) − v(µ) e1 (−¯

we get in combination with (3.17) v(µ)e− v ′ (µ) e− 2 (µ) 1 (µ) . = −i −1 e1 (−¯ µ) v (µ)e2 (−¯ µ) v(µ)s(µ)

¯ −λ ¯ 0 ), we have e2 (−¯ By the symmetry kˆ{s− ,ν− } (λ, λ0 ) = kˆ{s− ,ν− } (−λ, µ) = e1 (µ). Thus (3.16) is proved.


305

Corollary 3.5. Let µ ∈ R+ and as before Re λ0 > 0, then |e2 (µ)|2 − |e1 (µ)|2 =

1 (log v(µ))′ . i

(3.18)

Proof. All terms in (3.16) have boundary values. Recall that on the real axis (se− µ). Then use again the symmetry of the repro1,2 )(µ) = (s− e1,2 )(µ) − e1,2 (−¯ ducing kernel. We finish this section with a translation of the relation ; ; ; f ;2 ; f 2{s+ ,ν+ } = ; ; ve2 ; Kθ

2 ((3.15) is a unitary map from H{s to Kθ ) to the following proposition. + ,ν+ }

Theorem 3.6. Let sθ+ (λ) :=

¯ e2 (−λ) , e2 (λ)

λ ∈ R+ ,

(3.19)

¯ = sθ (λ) to the whole R. Let ν θ be a positive extended by the symmetry sθ+ (−λ) + + measure on the imaginary half-axis θ (λ) := dν+

|dv(λ)| , 2π|e2 (λ)|2

λ ∈ iR+ .

(3.20)

Then ||f ||2{s+ ,ν+ } + 1 θ |f (λ)|2 dν+ (λ) + = f (λ) 4π R iR+

¯ −f (−λ)

,#

1 sθ+ (λ)

sθ+ (λ) 1

$#

(3.21) $ f (λ) dλ ¯ −f (−λ)

2 for all f ∈ H{s . In other words + ,ν+ } 2 2 → H{s id : H{s θ ,ν θ } + ,ν+ } +

+

is an isometry. Proof. We use definition of the scalar product in Kθ , relations (3.13), (3.15), and (3.18).

4. Unitary node, II: a canonical system In this section we associate a canonical system (see [5, 18]) with the given chain 2 {H{s (x)}x∈R of subspaces of L2{s+ ,ν+ } . + ,ν+ }

306


4.1. Characteristic function of a unitary node and transfer matrix. Definitions This time we consider the unitary multiplication operator by v¯, v= with respect to the decomposition

λ2 − λ20 ¯2 , λ2 − λ 0

+ 2 ˆ2 L2{s+ ,ν+ } = (H {s− ,ν− } ) ⊕ K{s+ ,ν+ } (x) ⊕ H{s+ ,ν+ } (x).

(4.1)

Actually this is definition of the space K{s+ ,ν+ } (x). The following lemma is similar to Lemma 3.1. Lemma 4.1. The multiplication operator by v¯ acts from + (λ, λ0 )} ⊕ K{s+ ,ν+ } (x) ⊕ {k{s+ ,ν+ } (λ, λ0 ; x)} {kˆ{s − ,ν− }

(4.2)

+ ¯ 0 )} ⊕ K{s ,ν } (x) ⊕ {k{s ,ν } (λ, −λ ¯ 0 ; x)}. (λ, −λ {kˆ{s + + + + − ,ν− }

(4.3)

to We define normalized vectors that form orthonormal bases in E1 and E2 + (λ, λ0 ) kˆ{s k{s+ ,ν+ } (λ, λ0 ; x) (1) (1) − ,ν− } ; (4.4) e1 (λ) = , e2 (λ) = ||k{s+ ,ν+ } (λ, λ0 ; x)|| ||kˆ + (λ, λ0 )|| {s− ,ν− }

and (2)

e1 (λ) =

+ ¯0 ) ¯ 0 ; x) (λ, −λ kˆ{s k{s+ ,ν+ } (λ, −λ (2) − ,ν− } , e (λ) = + ¯ 0 ; x)|| . ¯ 0 )|| 2 ||k (λ, − λ ||kˆ{s (λ, − λ {s ,ν } + + ,ν } −

(4.5)

−

(i)

(i)

We point out that the vectors e2 (λ), i = 1, 2, depend on x and e1 (λ), i = 1, 2, do not. ˜1 ⊕ H ˜ 2 its Potapov-Ginzburg Generally for an operator A : H1 ⊕ H2 → H ˜2 → H ˜ 1 ⊕ H2 is defined by [15, 10] transform A˜ : H1 ⊕ H # $ # $ # $ # $ x y1 x y1 = A˜ 1 , where =A 1 . x2 y2 y2 x2 $ # A11 A12 we have In terms of the block decomposition of A = A21 A22 $# $ # # $# $ x1 A11 0 I −A12 y1 . = 0 −A22 x2 A21 −I y2 Therefore,

# I A˜ = 0

−A12 −A22

$−1 #

A11 A21

$ # A11 − A12 A−1 0 22 A21 = −I −A−1 22 A21

$ A12 A−1 22 . −1 A22

(4.6)

The transformation is well defined if A22 is invertible. Note, that if A is unitary, y1 2 + y2 2 = x1 2 + x2 2 ,


307

then A˜ preserves the indefinite metric y1 2 − x2 2 = x1 2 − y2 2 .

For the unitary node given by the multiplication operator by v¯ and decompositions (4.2), (4.3): (1)

(1)

(2)

(2)

U : (K ⊕ {e1 }) ⊕ {e2 } → (K ⊕ {e2 }) ⊕ {e1 }

(4.7)

˜ : (K ⊕ {e(1) }) ⊕ {e(2) } → (K ⊕ {e(2) }) ⊕ {e(1) } U 1 1 2 2

(4.8)

Θ(ζ) := PE2 U (I − ζPK U )−1 |E1 ,

(4.9)

we define the j-unitary node

by (4.6), separating in this way the x-depending “channels”. The characteristic operator-valued function for the node (4.7) is and its matrix with respect to the chosen bases is # $ + , #c $ + , c1 1 (1) (2) (2) Θ(ζ) e(1) θ(ζ) = , e (λ) e (λ) (λ) e (λ) 1 2 2 1 c c 2

2

where

#

$ θ11 θ12 (ζ). θ21 θ22 Respectively, its functional representation is of the form , #F $ + 1 (2) (ζ), PE2 U (I − ζPK U )−1 f = e(2) 2 (λ) e1 (λ) F θ(ζ) =

(4.10)

2

(4.11)

for f ∈ K{s+ ,ν+ } (x). The transfer matrix is actually the characteristic matrix function of the node (4.8). Having (4.9), (4.10), we rewrite (4.7) in the block form as ⎡ ⎤ ⎡ ⎤ ζk(ζ) k(ζ) # $ # $ # $ ⎣ ⎦ ⎣ c1 c ⎦. U = θ11 θ12 (4.12) (ζ) 1 c2 θ21 θ22 c2 ˜: Consequently, we get for U ⎤ ⎡ ⎤ ⎡ ζk(ζ) $ # $ k(ζ) $ # $ # # ˜⎣ c1 ⎦ = ⎣ θ11 (ζ) θ12 (ζ) c1 ⎦ . 1 0 U 0 1 c2 θ21 (ζ) θ22 (ζ) c2

(4.13)

Therefore the transfer matrix A(ζ) of the j-node is related to θ(ζ) by $ # $ # θ (ζ) θ12 (ζ) 1 0 = 11 . A(ζ) 0 1 θ21 (ζ) θ22 (ζ)

Thus

A(ζ) =

#

θ11 (ζ) 0

θ12 (ζ) 1

$#

1 θ21 (ζ)

0 θ22 (ζ)

$−1

=

#

1 −θ12 (ζ) 0 −θ22 (ζ)

$−1 #

θ11 (ζ) θ21 (ζ)

$ 0 . −1 (4.14)

308


4.2. Calculating θ and A We are following the same lines as in Section 3. Let us substitute (4.10) into (3.8) , + , + (2) (1) θ(ζ) = e(1) v(λ) e(2) 2 (λ) e1 (λ) 1 (λ) e2 (λ) ,. + (1) +(ζ − v(λ)) PK U (I − ζPK U )−1 e(1) (λ). e2 1

All terms here are analytic in λ and we can choose λ ∈ C+ with the property v(λ) = ζ. Then we get , , + + (1) (2) . (4.15) θ(v(λ)) = e(1) v(λ) e(2) 1 (λ) e2 (λ) 2 (λ) e1 (λ) Similarly, by (4.11)

+ v(λ) e(2) 2 (λ)

(2)

e1 (λ)

,#

$ F1 (v(λ)) = f (λ). F2

(4.16)

¯ λ ∈ R, in (4.15), It is tempting to make the change of variable λ → −λ, (4.16) and to write + , + , (2) − (1) − − − v(λ) (e(2) θ(v(λ)) = (e(1) , (4.17) 2 ) (λ) (e1 ) (λ) 1 ) (λ) (e2 ) (λ) and

+ − v(λ) (e(2) 2 ) (λ)

(2)

(e1 )− (λ)

, #F $ 1

F2

(v(λ)) = f − (λ).

(4.18)

However, to succeed with this plan, we need to prove that θ(ζ) has an analytic continuation in C \ D etc. That is why we prefer to consider a dual node given by the diagram K⊕ ⏐ E1 ⏐ F– −

K ⊕

E1−

v ¯

−→ v ¯

K⊕ ⏐ E2 ⏐ F– −

−→ K ⊕

(4.19)

E2−

The characteristic matrix-valued function remains the same since we choose basis − as the image of the basis in E1,2 . Then we obtain (4.17) and (4.18) simply in E1,2 repeating the arguments from (4.15) and (4.16). Hence : : 9 9 # $ (1) (1) (2) (2) θ (e1 )− (e2 )− θ12 (e2 )− (e1 )− (λ), (4.20) (λ) 11 (v(λ)) = v(λ) (1) (1) (2) (2) θ21 θ22 e1 e2 e2 e1 and v(λ)

9

(2)

(e2 )− (2) e2

: # $ # −$ (2) F1 f (e1 )− (λ). (λ) (v(λ)) = (2) F2 f e1

(4.21)


Lemma 4.2. Both det

9 (2) (e2 )− (2) e2

Furthermore, θ22 (λ) det

9

309

: (2) (e1 )− (λ) and θ22 (λ) do not vanish identically. (2) e1

(2)

(e2 )− (2) e2

:

′ (2) 1 i (e1 )− (λ) = − . (2) s(λ) v(λ) e1

(4.22)

In particular, the characteristic matrix-valued function θ(ζ) and the map K{s+ ,ν+ } (x) → Kθ are well defined by (4.20), (4.21) in terms of the reproducing kernels. Proof. This follows from an obvious consequence of (4.20) : : 9 9 # $ (2) (2) (2) (1) 1 −θ12 v(e2 )− −(e2 )− (e2 )− (e1 )− (λ) (λ), (4.23) (v(λ)) = v(λ) (2) (2) (2) (1) 0 −θ22 e2 e1 ve2 −e2 and (3.16) that says

9

(2)

v(e2 )− det (2) ve2

: (1) v ′ (λ) −(e2 )− . (λ) = i (1) s(λ) −e2

By (4.14), we have for the transfer matrix $−1 # # θ11 1 −θ12 Ax (λ2 ) = θ21 0 −θ22

$ 0 (v(λ)). −1

(4.24)

The map from Kθ to the corresponding de Branges space H(A) [5, Sect. 28] is of the form # $ $−1 # $ # A1 F1 1 −θ12 (λ2 ) = (v(λ)). (4.25) A2 0 −θ22 F2 Combining (4.20), (4.24) with : 9 # (2) (2) θ11 (e2 )− (e1 )− (λ) v(λ) (2) (2) θ21 e2 e1

we have 9 and

(2)

v(e2 )− (2) ve2 9

9 $ (1) (e1 )− 0 (v(λ)) = (1) −1 e1

: 9 (1) (1) −(e2 )− (e1 )− 2 (λ)Ax (λ ) = (1) (1) −e2 e1 (2)

v(e2 )− (2) ve2

: (2) −v(e1 )− (λ), (4.26) (2) −ve1

: (2) −v(e1 )− (λ), (2) −ve1

: # $ # −$ (1) A1 −(e2 )− f (λ). (λ) (λ2 ) = (1) A2 f −e2

Note that the condition I − θθ∗ ≥ 0 is the same as $# $# $# # $# $∗ # θ11 1 −θ12 1 0 1 −θ12 1 0 0 θ − 11 θ21 −1 0 −1 θ21 0 −θ22 0 −1 0 −θ22

(4.27)

(4.28) $∗ 0 ≥ 0. −1

310


That is, the transformation (4.24) maps contractive matrices onto j-contractions, $ # 1 0 , j − AjA∗ ≥ 0, j := 0 −1 although this was clear from the definition of j-node (4.8). 4.3. de Branges’ Theorem Theorem 4.3. For every x > 0, : 9 9 (1) (1) −is(λ) −e(1) (e1 )− (e2 )− 2 Ax (λ2 ) = ′ (λ) (1) (2) − (2) v (λ) −ve2 e1 v(e2 )

: (2) −v(e1 )− (λ) (2) −ve1

(4.29)

is an entire matrix-valued function of λ2 and 9 : # $ /# 2 $ (1) −is(λ) −e(1) f− (e2 )− A1 2 (λ), f ∈ K H(Ax ) = (λ2 ) = ′ (λ) (x) , {s+ ,ν+ } (2) − A2 f v (λ) −ve(2) 2 v(e2 ) (4.30) is the de Branges space of entire functions [5, Sect. 28]. Proof. The entries of the matrix-valued function in RHS of (4.29) are holomorphic in the z-plane, z = λ2 , cut along the positive half-axis. Moreover, the limits Ax (ξ + i0) and Ax (ξ − i0) coincide for a.e. ξ ∈ R+ . We have to prove analyticity of the matrix-valued function on the cut. Fix ξ0 ∈ R+ and a disk√Dξ0 = D(ξ0 , r) of a sufficiently small radius r centered at the point ξ0 , such that |s( z)| is bounded from above and below on its boundary. This is possible because s has non-vanishing boundary values almost everywhere. We would like to apply Morera’s Theorem [9, p. 95]. However, the glued functions, generally speaking, do not belong to H 1 in both half-disks {z ∈ Dξ0 : Im z > 0} and {z ∈ Dξ0 : Im z < 0}. For this reason, first, we have to use the normalization ˆ2 procedure given in Lemma 2.6. Recall that for s+ < 1 every function from H belongs to the standard H 2 even without the factor s. Hence, for each ǫ we get that the matrix-valued function Ax (z; ǫ) is holomorphic on Dξ√ 0 . On the boundary of this disk Ax (z; ǫ) goes to Ax (z) in L1 with the weight |s( z)||(z − ξ0 )2 − r2 |, that is, actually, in L1 after multiplication by (z − ξ0 )2 − r2 . Therefore, Ax (z) is also holomorphic on the disk. We point out that for all x the x-depending matrix in the RHS of (4.29) meets the following normalization condition (2)

(ve2 )(λ0 ) = 0,

(1)

e2 (λ0 ) > 0.

(4.31)

As the result we get a family of 2 × 2 j-contractive matrix-valued functions with a certain normalization at λ0 . The family is monotonic in x, and every matrix is an entire function in λ2 of the zero mean type (concerning the corollary of the


311

last condition see [5], Theorem 39). According to de Branges’ Theorem [5, Sect. 36, 37], Theorem 37, such a family can be included in the chain $ ? $ # # d 0 −β α β (t) A(λ2 , t), |β| = α, (t) + ¯ j A(λ2 , t) = iλ2 ¯ (4.32) β 0 β α dt such that Ax (λ2 ) = A(λ2 , tx ), where x = x(t) is a monotonic function. Here we choose λ20 = i as the normalization point. 4.4. Parameters of the system in terms of reproducing kernels Theorem 4.4. For the system (4.32) ¯0) ˆ 0 , −λ s(λ0 ) d k(λ β =− . ¯ ˆ ¯ ¯0) α s(−λ0 ) d k(−λ0 , −λ

(4.33)

Proof. Set E=

9

(1)

−e2 (2) −ve2

: # (1) −kλ0 1 (e2 )− (2) − = v(e2 ) k(λ0 , λ0 ) −vk−λ¯ 0

$ kλ−0 . − vk−λ¯ 0

(4.34)

The subscripts of the kernels are obvious (see (4.4), (4.5)) and are omitted. We ˙ −1 = j EE ˙ −1 . In particular, for λ2 = i note that due to (4.29) j AA 0 $ # ˙ 0 )E −1 (λ0 ) = − α 2β . (4.35) j E(λ 0 α On the other hand, using (2.3), (2.4), we have $ $# # −1 0 τ aτ E(λ0 ) = −1 0 C 0 τ with a constant C and τ := k(λ0 , λ0 ),

ˆ 0 , −λ ¯ 0 ). a := s2 (λ0 )(2Im λ0 )2 k(λ

In these notations,

˙ 0 )E −1 (λ0 ) = j E(λ Comparing this with (4.35) we get

β α

# τ˙ τ

0

aτ ˙ 2 τ˙ τ

$

(4.36)

(4.37)

.

= − d(τda−2 ) . Since

¯ 0 )(2Im λ0 )2 k(− ¯ 0 ), ˆ λ ¯ 0 , −λ τ −2 = s(λ0 )s(−λ we come to (4.33) using (4.37)

312


5. de Branges system and Sturm-Liouville equation In this section, we rewrite the results of the previous sections for a particular case of the Sturm-Liouville equation. Let Ly = −y ′′ + qy

(5.1)

be a self-adjoint operator acting on L2 (R), and let ¯2 L−λ 0 , Re λ0 > 0, Im λ0 > 0, u= L − λ20

(5.2)

be its Cayley transform.

Lemma 5.1. Let e± (x, λ) ∈ L2 (R± ), e± (x, λ) = 1, be such that d2 ± e (x, λ)+q(x)e± (x, λ) = λ2 e± (x, λ), dx2 Then for all f+ ∈ L2 (R+ ) −

x ∈ R± , Im λ > 0, Re λ = 0. (5.3)

¯ 0 ) + g+ , uf+ = Ce− (x, λ0 )f+ , e+ (x, −λ

(5.4)

2

where C = C(u) and g+ ∈ L (R+ ). Proof. Let

(uf+ )(x) =

/

g− (x), g+ (x),

x ∈ R− , x ∈ R+

or, what is the same, / 2 ¯2 g− (x), λ0 − λ0 (x) = f + L − λ20 g+ (x) − f+ (x),

x ∈ R− . x ∈ R+

Therefore, ′′ −g− + qg− = λ20 g− ,

and g− ∈ L2 (R− ). That is,

x ∈ R− ,

g− (x) = C(f+ )e− (x, λ0 ).

Thus we get uf+ = C(f+ )e− (x, λ0 ) + g+ . For C(f+ ) we have C(f+ ) = uf+ , e− (x, λ0 ) = f+ , u∗ e− (x, λ0 ) .

Now we are looking at ∗ −

u e (x, λ0 ) = or

¯ 2 − λ2 λ I + 0 ¯ 20 e− (x, λ0 ) = L − λ0

¯ 2 − λ2 )e− (x, λ0 ) = (λ 0 0

/

/

h− (x), x ∈ R− h+ (x), x ∈ R+

˜ ˜− − λ ¯2h ˜ ′′ + q h −h − 0 − , x ∈ R− ˜ + , x ∈ R+ ˜ ′′ + q h ˜+ − λ ¯2h −h + 0

,

,

Inverse Scattering Problem and Canonical Systems ˜ = h − e− (x, λ0 ). This implies where h / ¯ 0 ), x ∈ R− −e− (x, λ0 ) + C1 e− (x, −λ ˜ h= ¯ 0 ), x ∈ R+ C2 e+ (x, −λ

313

,

where C1 , C2 are defined by the conditions ˜ ˜ ˜ ′ (−0) = h ˜ ′ (+0). h(−0) = h(+0), h Notice that the equality C2 = 0 contradicts the linear independence of e− (x, λ0 ) ¯ 0 ). and e− (x, −λ Hence, / − ¯ ˜ + e− (x, λ0 ) = C1 e (x, −λ0 ), x ∈ R− , u∗ e− (x, λ0 ) = h (5.5) ¯ 0 ), x ∈ R+ C2 e+ (x, −λ and (5.4) is proved with ¯ 0 ) . C(f+ ) = f+ , C2 e+ (x, −λ

¯ 0 )} to L (R+ ) ⊕ Corollary 5.2. The operator u acts from L (R+ ) ⊕ {e (x, −λ {e− (x, λ0 )}. 2

−

2

Proof. Similarly to (5.4), ¯ 0 ) + g− . uf− = Ce+ (x, λ0 )f− , e− (x, −λ (5.6) 2 − ¯ That is, uf− = g− ∈ L (R− ) if f− ⊥ e (x, −λ0 ). Moreover, (uf− ) is orthogonal to e− (x, λ0 ) by (5.5) in this case. Corollary 5.3. For every x0 > 0, the operator u acts from ¯ 0 )}) ⊕ {e+ (x, −λ ¯ 0 )} (L2 [0, x0 ] ⊕ {e− (x, −λ x0

(5.7)

to

where

e+ x0 (x, λ)

2

− (L2 [0, x0 ] ⊕ {e+ x0 (x, λ0 )}) ⊕ {e (x, λ0 )}

(5.8)

∈ L [x0 , ∞) is the normalized solution of (5.3).

Below, the dot means the derivative with respect to x. Theorem 5.4. The transfer matrix of the unitary node (5.7), (5.8) is of the form $ $ # − # + ¯ 0 ) e− (0, λ0 ) ¯ −1 −e (0, −λ ex0 (x0 , λ0 ) −e+ 2 x0 (x0 , −λ0 ) Ax0 (λ2 ) = + B (λ ) x 0 ¯ ¯ 0 ) e˙ − (0, λ0 ) , −e˙ − (0, −λ e˙ x0 (x0 , λ0 ) −e˙ + x0 (x0 , −λ0 ) (5.9) where # $ c(x, λ) s(x, λ) Bx (λ2 ) = (5.10) c(x, ˙ λ) s(x, ˙ λ) is the standard transfer matrix for equation (5.1) $ # d 0 1 Bx (λ2 ), B0 (λ2 ) = I. Bx (λ2 ) = (5.11) 2 q(x) − λ 0 dx

314


Proof. In the block form we have ⎡ ⎤ ⎤ ⎡ #kζ $ #ζkζ$ u ⎣ c1 ⎦ = ⎣ d1 ⎦ , c2 d2 with

# $ # $ c d1 = A(ζ) 1 . d2 c2

(5.12)

(5.13)

In other words, ¯2 λ2 − λ 0 ¯ 0 ) + d2 e+ (x, −λ ¯ 0 )} I+ 0 {ζkζ + c1 e− (x, −λ x0 L − λ20 − =kζ + d1 e+ x0 (x, λ0 ) + c2 e (x, λ0 ),

or ¯2 λ20 − λ 0 ¯ 0 ) + d2 e+ (x, −λ ¯ 0 )} {ζkζ + c1 e− (x, −λ x0 L − λ20

− − + ¯ ¯ =(1 − ζ)kζ + d1 e+ x0 (x, λ0 ) + c2 e (x, λ0 ) − c1 e (x, −λ0 ) − d2 ex0 (x, −λ0 ). (5.14)

This means that the RHS of (5.14) has the second derivative and we have on the interval [0, x0 ] (5.15) −kζ′′ + qkζ = λ2 kζ . for the spectral parameter

λ2 = λ20 + Above, ζ = Let

λ2 − λ20 ¯2 . λ2 − λ 0

ζ ¯ 2 ). (λ2 − λ 0 1−ζ 0

(1 − ζ)kζ = Ac(x, λ) + Bs(x, λ).

(5.16)

Then the continuity at x = 0 implies $# $ # $ # − ¯ 0 ) e− (0, λ0 ) c1 A −e (0, −λ = ¯ 0 ) e˙ − (0, λ0 ) c2 , −e˙ − (0, −λ B

and by the continuity at x = x0 , $# $ # $# $ # + ¯ d1 ex0 (x0 , λ0 ) −e+ c(x0 , λ) s(x0 , λ) A x0 (x0 , −λ0 ) . = + ¯ d2 ˙ 0 , λ) B e˙ + c(x ˙ 0 , λ) s(x x0 (x0 , λ0 ) −e˙ x0 (x0 , −λ0 ) The theorem is proved.

We now compute the parameters of the related canonical system under the chosen normalization. We start observing that, up to the initial matrix A0 , the transfer matrix has the same normalization as the transfer matrix (4.24) (or (4.29)) in Section 4.


315

Corollary 5.5. The transfer matrix of unitary node (5.7), (5.8) is of the form ˜ x (λ2 )A0 , Ax (λ2 ) = A where

$ $ # + # + + ¯ ¯ −1 e0 (0, λ0 ) −e+ 0 (0, −λ0 ) . ˜ x (λ2 ) = ex+0 (x0 , λ0 ) −ex+0 (x0 , −λ0 ) A Bx0 (λ2 ) + + 0 ¯ ¯ e˙ x0 (x0 , λ0 ) −e˙ x0 (x0 , −λ0 ) e˙ 0 (0, λ0 ) −e˙ 0 (0, −λ0 ) (5.17) ˜ x (λ2 ) meets the normalization: Therefore, A ˜ x (λ2 ))21 = 0 ˜ x (λ2 ))11 > 0, (A (A 0

0

for all x > 0. We use the same notation as in (4.36). Theorem 5.6. Let m+ (λ) be the Weyl function of operator (5.1) and let $ # ˜ x (λ2 ) = τ aτ A −1 . 0 0 τ

(5.18)

Then

da c(x, λ0 ) m+ (λ0 ) + s(x, λ0 ) = . d(τ −2 ) c(x, λ0 ) m+ (λ0 ) + s(x, λ0 ) Proof. First of all,

#

$ $ # e+ m+ (λ) x0 (x, λ) = Bx ρ(x0 ), + 1 e˙ x0 (x, λ)

x ≥ x0 ,

(5.19)

(5.20)

where ρ(x0 ) should be found from the condition ∞ 2 |e+ x0 (x, λ)| dx = 1. x0

Using

# $ d ¯ 2 )B∗ (λ2 ) 1 1 0 Bx (λ2 ), {B∗x (λ2 )JBx (λ2 )} = −(λ2 − λ x 0 dx $ # 0 1 , we obtain with J = −1 0 $ # # $ ∞ m+ (λ) 1 1 0 Bx (λ2 ) ρ2 (x0 ) dx m+ (λ) 1 B∗x (λ2 ) 1 0 x0 $ # m+ (λ) m+ (λ) 1 B∗x0 (λ2 )JBx0 (λ2 ) 1 2 = ρ (x0 ) . ¯2 λ2 − λ That is, ¯2 λ2 − λ $. # ρ2 (x0 ) = m+ (λ) m+ (λ) 1 B∗x0 (λ2 )JBx0 (λ2 ) 1

(5.21)

(5.22)

(5.23)

316


In particular ρ2 (0) = −

¯2 λ2 − λ

m+ (λ) − m+ (λ)

.

Since for x ≥ x0 $ # + $ $ # # + ex0 (x, λ0 ) ρ(0) m+ (λ0 ) e0 (0, λ0 ) ρ(0) = + = Bx (λ20 ) , Bx (λ20 ) + 1 e˙ x0 (x, λ0 ) ρ(x0 ) e˙ 0 (0, λ0 )

(5.24)

(5.25)

˜ x (λ2 ) (5.17) we get for the first column of the matrix A 0 $ $ # # + # $ −1 + + ¯ ex0 (x0 , λ0 ) ρ(0) ex0 (x0 , λ0 ) −ex0 (x0 , −λ0 ) 1 ρ(0) = . + ¯ 0 ρ(x0 ) e˙ + e˙ + x0 (x0 , λ0 ) ρ(x0 ) x0 (x0 , λ0 ) −e˙ x0 (x0 , −λ0 ) ρ(0) and, recalling (5.23), (5.24), we Therefore, we deduce from (5.18) that τ = ρ(x 0) come to m+ (λ0 ) − m+ (λ0 ) $. # (5.26) τ −2 = − m+ (λ0 ) m+ (λ0 ) 1 B∗x0 (λ20 )JBx0 (λ20 ) 1

To compute aτ , we proceed as

# $ m+ (λ0 ) 2 ¯ e+ ρ(0) x0 (x0 , −λ0 ) Bx0 (λ0 ) 1 $ # m+ (λ0 ) 2 + ¯ ¯ ρ(0) = − e+ x0 (x0 , −λ0 ) e˙ x0 (x0 , −λ0 ) JBx0 (λ0 ) 1 $ # m+ (λ0 ) ρ(0), = − ρ(x0 ) m+ (λ0 ) 1 B∗x0 (λ20 )JBx0 (λ20 ) 1

¯ aτ ∆ = − −e˙ + x0 (x0 , −λ0 )

where

#

$ + ¯ e+ x0 (x0 , λ0 ) −ex0 (x0 , −λ0 ) + + ¯0 ) e˙ x0 (x0 , λ0 ) −e˙ x0 (x0 , −λ $ # + + x0 (x0 , λ0 ) ¯ 0 ) J e+ ¯ 0 ) e˙ + (x0 , −λ = ex0 (x0 , −λ x0 e˙ x0 (x0 , λ0 ) $ # ∗ 2 m+ (λ0 ) ρ(x0 ). =ρ(x0 ) m+ (λ0 ) 1 Bx0 (λ0 )JBx0 (λ20 ) 1

∆ = det

(5.27)

(5.28)

Combining (5.27) and (5.28), we obtain

a=−

m+ (λ0 ) m+ (λ0 )

$ # m+ (λ0 ) 1 B∗x0 (λ20 )JBx0 (λ20 ) 1 $. # m+ (λ0 ) 2 2 ∗ 1 Bx0 (λ0 )JBx0 (λ0 ) 1

(5.29)

Using (5.21) and the Wronskian identity for Bx (λ) we get (5.19) from (5.26) and (5.29) by a direct computation.


317

6. Appendix 1. An example In this section we give an example which shows that the class of canonical systems discussed in Section 4 is larger than the class of Sturm-Liouville equations from Section 5. We will see that generally 2 ˆ2 H{s = H {s+ ,ν+ } , + ,ν+ } 2 H{s + ,ν+ }

(6.1)

ˆ2 H {s+ ,ν+ }

although always and we will also discuss some other inter⊂ esting phenomena. λ and ν+ = 0. Throughout this section we set s+ = λ+i First, we prove (6.1). Since ¯ =0 (s+ f )(λ), −f (−λ) 2 for all f ∈ H 2 , we get ||f ||{s+ ,ν+ } = ||f ||. Therefore, in this case H{s coincides + ,ν+ } with the standard H 2 . i , so s · 1 ∈ H 2 . Let us check that On the other hand, we have s(λ) = λ+i 1 ∈ L2{s+ ,ν+ } . This follows from the identity 9 :# $ λ 1 2 1 λ+i 1 −1 = . (6.2) λ −1 |λ + i|2 1 λ+i

ˆ2 Hence, by the definition of H {s+ ,ν+ } a constant function belongs to this space, but of course 1 ∈ H 2 and (6.1) is proved. The above conclusion can be sharpened. Using $B $# $ # A# 1 f (λ) 1 1 s+ (λ) ¯ = 0, = s+ (λ) − 1, −f (−λ) , (6.3) ¯ −f (−λ) −1 1 s+ (λ) 2

2 ˆ2 for all f ∈ H 2 , we get that 1 is orthogonal to H{s ⊂H {s+ ,ν+ } . Actually we + ,ν+ } have the following orthogonal decomposition 2 2 ˆ2 H {s+ ,ν+ } = {1} ⊕ H{s+ ,ν+ } = {1} ⊕ H .

(6.4)

ˆ2 This implies that the reproducing kernel of H {s+ ,ν+ } is ˆ λ0 ) = kˆ{s ,ν } (λ, λ0 ) = k(λ, + +

1 i + ¯0 , ||1||2{s+ ,ν+ } λ−λ

(6.5)

and, by (6.2), ||1||2{s+ ,ν+ } = 12 .

2 ˆ2 Now we show that the property H{s (x) = H {s+ ,ν+ } (x) is not x-invariant. + ,ν+ } 2 2 ˆ (x) for x > 0 despite (6.1) for x = 0. (x) = H Namely, H {s+ ,ν+ }

Lemma 6.1. Let s+ =

{s+ ,ν+ }

λ λ+i ,

2 ˆ2 ν+ = 0. Then H{s (x) = H {s+ ,ν+ } (x) for all x > 0. + ,ν+ }

318


Proof. Notice that iλx ˆ 2 ˆ2 H H{s+ e2iλx ,ν+ e2iλx } , {s+ ,ν+ } (x) = e 2 2 ˆ 2 2iλx (x) = eiλx H 2 . So we have to show that H and H{s {s+ e ,ν+ e2iλx } = H . + ,ν+ } 2 ˆ By definition f ∈ H{s+ e2iλx ,ν+ e2iλx } means that

i f (λ) ∈ H 2 , λ+i λ ¯ ∈ L2 . e2iλx f (λ) − f (−λ) λ+i We have to prove that f ∈ H 2 . i Let g(λ) = λ+i f (λ). Conditions (6.6) can be easily transformed into

(6.6)

¯ ∈ L2 λ{e2iλx g(λ) + g(−λ)}

with g ∈ H 2 . Let G denote the Fourier transform of g. Obviously, G ∈ L2 (R+ ) since g ∈ H 2 . In these terms we have {G(2x + t) + G(−t)}′ ∈ L2 .

(6.7)

Since the supports of the functions G(2x + t) and G(−t) do not intersect, we get from (6.7) that G′ (t) ∈ L2 . Therefore λg(λ) ∈ L2 , and, consequently, f belongs to L2 and, in fact, to H 2 . Corollary 6.2. Let s+ = x < 0.

λ λ+i ,

2 ˆ2 (x) = H ν+ = 0. Then H{s {s+ ,ν+ } (x) for all + ,ν+ }

Proof. We only have to mention that s− = s+ in our case and to use Theorem 1.3. Corollary 6.3. Let s+ =

λ λ+i ,

ν+ = 0. Then

2 ˆ2 lim H{s (x) = H {s+ ,ν+ } (0). + ,ν+ }

x→−0

(6.8)

2 2 (x0 ) for x0 ≥ 0. Therefore by (x) = H{s Proof. Obviously, limx→x0 +0 H{s + ,ν+ } + ,ν+ } s− = s+ and the duality stated in Theorem 1.3,

lim

x→x0 −0

ˆ2 ˆ2 H {s+ ,ν+ } (x) = H{s+ ,ν+ } (x0 )

for x0 ≤ 0. Finally, we use Corollary 6.2

2 ˆ2 ˆ2 lim H{s (x) = lim H {s+ ,ν+ } (x) = H{s+ ,ν+ } (0). + ,ν+ }

x→−0

x→−0

This means, in particular, that the canonical system related to the given scattering data is not a Sturm-Liouville equation. Indeed, let A(x0 , x1 ; λ2 ), x0 < x1 , be the transfer matrix (4.24) (or (4.29)). Recalling (4.27), we write + , + , (2) (1) (1) (2) 2 ve2 (x1 ) e2 (x1 ) = e1 (x0 ) ve1 (x0 ) A(x0 , x1 ; λ ).

Inverse Scattering Problem and Canonical Systems We also introduce B(x0 , x1 ; λ2 ) by , + , + (2) (1) (2) (1) 2 ve2 (x1 ) e2 (x1 ) = ve2 (x0 ) e2 (x0 ) B(x0 , x1 ; λ ),

319

(6.9)

so that we have the chain rule

A(x0 , x2 ; λ2 ) = A(x0 , x1 ; λ2 )B(x1 , x2 ; λ2 ). Fix x0 < 0, put x2 = 0, and let B(λ2 ) = lim B(x1 , 0; λ2 ). x1 →−0

Using (6.9) and (6.8) we have + , + , (2) (1) (2) (1) 2 ve2 (x1 ) e2 (x1 ) = vˆ e2 (x1 ) eˆ2 (x1 ) B(λ ),

(6.10)

where (see (4.4), (4.5)) (1)

eˆ2 (λ) =

kˆ{s+ ,ν+ } (λ, λ0 ; x) , ||kˆ{s ,ν } (λ, λ0 ; x)|| +

(2)

eˆ2 (λ) =

+

¯ 0 ; x) kˆ{s+ ,ν+ } (λ, −λ . ˆ ¯ 0 ; x)|| ||k{s ,ν } (λ, −λ +

+

Thus B(λ2 ) is a non-trivial divisor of A(x0 , x; λ2 ) for x ≥ 0. The chain A(x0 , x; λ2 ) is not even continuous in x, lim A(x0 , x; λ2 ) = A(x0 , 0; λ2 ).

x→−0

Of course, we can get an explicit formula for B(λ2 ). Since B depends on λ2 , (6.10) implies :# :# 9 $ 9 (2) $ (2) (1) (1) v(λ) 0 eˆ2 (λ) eˆ2 (λ) e2 (λ) e2 (λ, λ0 ) v(λ) 0 = B(λ2 ), (1) (2) (1) (2) 0 1 0 1 e (−λ) e2 (−λ) eˆ2 (−λ) eˆ2 (−λ) or, with the help of (6.5), > $ $# ˆ 0 , λ0 ) # k(λ, −λ ¯0) k(λ v(λ) 0 k(λ, λ0 ) ¯ 0 ) k(−λ, λ0 ) 0 1 k(λ0 , λ0 ) k(−λ, −λ # $# $ ˆ ¯ ˆ k(λ, −λ0 ) v(λ) 0 k(λ, λ0 ) = ˆ B(λ2 ) ˆ ¯ 0 ) k(−λ, 0 1 λ0 ) k(−λ, −λ $ # $? # # ¯0 ) k(λ, −λ v(λ) k(λ, λ0 ) 1 1 + = 2 ¯ 0 k(−λ, −λ0 ) k(−λ, λ0 ) 1 1 Thus, directly, 1 + 4Im λ0 I =

i Re λ0

#

λ2 − λ20 −λ2 + λ20

(6.11) $ 0 B(λ2 ). 1

$ ? ¯2 λ2 − λ 2 0 2 2 + I B(λ ). ¯ −λ + λ0

Note that the determinant of the matrix in curly brackets is 1 + 4Im λ0 , so B(λ2 ) is indeed an entire function of λ2 (a linear polynomial), $? # 2 ¯ 2 −λ2 + λ ¯2 i 1 −λ + λ 0 0 . I+ B(λ2 ) = √ 2 2 2 2 λ − λ0 Re λ0 λ − λ0 1 + 4Im λ0

320


7. Appendix 2. On a certain sufficient condition 7.1. On a certain sufficient condition The following lemmas are related to attempts to rewrite the A2 condition for the spectral density [20] directly in terms of the scattering function. Lemma 7.1. Let W =

#

1 s+

$ s¯+ . 1

The following conditions are equivalent A # $ # $B A # $ # $B f (t) f (t) f (t) f (t) W −1 P+ W ¯ ¯ , P+ W ¯ ¯ ≤Q W ¯ ¯ , ¯ ¯ tf (t) tf (t) tf (t) tf (t) for all f ∈ L2s+ ⊖ Hs2+ and

ˆ 2 , t ∈ T. for f − (t) ∈ H s−

||f − ||2 ≤ Q||f − ||2s−

(7.1)

(7.2)

ˆ 2 is equivˆ 2 = H 2 , moreover the norm in H Lemma 7.2. If (7.2) holds, then H s− s− s− alent to the standard H 2 -norm. ˆ s2 = Hs2 ˆ s2 = Hs2 implies H Lemma 7.3. H + + − − ˆ 2 )+ ˆ 2 = (L2 ⊖ H 2 )+ = (L2 ⊖ H H s− s− s+ s− s−

Proof.

ˆ s2 )+ = Hs2 . = L2s+ ⊖ (H − +

Nevertheless, we cannot guarantee that the norm in Hs2+ is equivalent to the H 2 -norm. Thus in addition to (7.2) we have to impose the condition A # $ # $B f (t) f (t) f, f ≤ Q W ¯ ¯ , ¯ ¯ (7.3) tf (t) tf (t)

for all f ∈ H 2 (this is exactly the condition on equivalence of the norms). Obviously, the last inequality is the same as # $B A # # $ # $ $B A f (t) f (t) f (t) f (t) ≤Q W ¯ ¯ , ¯ ¯ , f ∈ H 2. (7.4) W P+ ¯ ¯ , P+ ¯ ¯ tf (t) tf (t) tf (t) tf (t) Thus we get

Theorem 7.4. The combination of the following two conditions 1

with F = W

1 2

#

1

1

1

(W − 2 P+ W 2 )F, (W − 2 P+ W 2 )F ≤ QF, F $ f (t) and f ∈ L2s+ ⊖ Hs2+ , and t¯f (t¯) 1

1

1

1

(7.5)

(W 2 P+ W − 2 )F, (W 2 P+ W − 2 )F ≤ QF, F (7.6) $ 1 f (t) 2 with F = W 2 ¯ ¯ and f ∈ H is equivalent to the first (or the second) condition tf (t) from [20], Theorem 3.1. #


321

Proof. Relation (7.5) is a slight modification of (7.1) and (7.6) of (7.4), respectively. Observe that if (7.6) holds with f ∈ H 2 , then every f ∈ Hs2+ belongs to H 2 . Therefore, in fact, (7.6) has a perfect sense for f ∈ Hs2+ . In any case, W ∈ A2 is a sufficient condition for (7.5), (7.6). Let us transform this matrix condition into a scalar one. Lemma 7.5. W is in A2 if and only if |s+ − s+ I |2 + (1 − |s+ I |2 ) 1 dm < ∞, sup 1 − |s+ |2 I |I| I where for an arc I ⊂ T we put

s+ I :=

1 |I|

(7.7)

s+ dm.

(7.8)

I

Proof. By definition we have that there exists Q > 0 such that W −1 I ≤ QW −1 I

for all I ⊂ T. Note that $ # # 1 s+ I 1 W I = = s+ I s+ I 1

0 1

$#

1 0

Therefore (7.9) is equivalent to # $ $# # 1 0 1 1 s+ I −1 W I s+ I 0 1 − |s+ I |2 0 1

(7.9)

0 1 − |s+ I |2

$#

1 0

$ s+ I . 1

$# $ 0 0 1 ≤ Q. 1 0 1 − |s+ I |2

Since the matrix in the RHS is positive, its boundedness is equivalent to the boundedness of its trace. The last condition with a small effort gives (7.7) and vice versa.

Acknowledgments. A part of this work was done during the last author’s visit to CMI (Centre de Mathématiques et Informatique) at University of Provence. He would like to thank the Department for the hospitality.

References [1] D. Arov, H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems. I. Foundations, Integral Equations Operator Theory, 29 (1997), no. 4, 373–454. [2] D. Arov, H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems. II. The inverse monodromy problem, Integral Equations Operator Theory, 36 (2000), no. 1, 11–70. [3] D. Arov, H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems. III. More on the inverse monodromy problem, Integral Equations Operator Theory, 36 (2000), no. 2, 127–181.

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[4] D. Arov, H. Dym, Strongly regular J-inner matrix-valued functions and inverse problems for canonical systems, Oper. Theory Adv. Appl., vol. 160, Birkhäuser, Basel, 2005, 101–160. [5] L. de Branges, Hilbert spaces of entire functions, Prentice-Hall, Englewood Cliffs, N.J. 1968. [6] M. Brodskii, Characteristic matrix-functions of linear operators, Mat. Sb., 39 (81) (1956), 176–200 (Russian). [7] M. Brodskii, M. Livshits, Spectral analysis of non-selfadjoint operators and open systems, Uspekhi Mat., 13 1 (79) (1958), 3–85 (Russian). [8] D. Damanik, R. Killip, B. Simon, Necessary and sufficient conditions in the spectral theory of Jacobi matrices and Schr¨ odinger operators, Int. Math. Res. Not. 2004, no. 22, 1087–1097. [9] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [10] V. Katsnelson, Methods of J-theory in continuous interpolation problems of analysis, Hokkaido University, Sapporo, 1985. [11] V. Katsnelson, A. Kheifets, P. Yuditskii, An abstract interpolation problem and the theory of extensions of isometric operators, Naukova Dumka, Kiev, 1987, 83–96 (Russian); translation in: Oper. Theory Adv. Appl., 95 (1997), 283–298. [12] A. Kheifets, Scattering matrices and Parseval equality in the abstract interpolation problem, Ph. D. thesis, Kharkov, 1989. [13] A. Kheifets, P. Yuditskii, An analysis and extension of V.P. Potapov’s approach to interpolation problems with applications to the generalized bi-tangential SchurNevanlinna-Pick problem and J-inner-outer factorization, Oper. Theory Adv. Appl., 72 (1994), 133–161. [14] V. Marchenko, Sturm-Liouville operators and applications, Birkh¨ auser Verlag, Basel, 1986. [15] V. Potapov, The multiplicative structure of J-contractive matrix functions, Amer. Math. Soc. Transl. (2), vol. 15 (1960), 131–243. [16] M. Reed, B. Simon, Methods of modern mathematical physics, I, Academic Press, New York, 1980. [17] M. Reed, B. Simon, Methods of modern mathematical physics, III, Academic Press, New York, 1979. [18] Ch. Remling, Schr¨ odinger operators and de Branges spaces, J. Funct. Anal., 196 (2002), no. 2, 323–394. [19] T. Tao, Ch. Thiele, Nonlinear Fourier Analysis, IAS/Park City Mathematics Series. [20] A. Volberg and P. Yuditskii, On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length, Comm. Math. Phys., 226 (2002), 567–605. S. Kupin CMI, Université de Provence 39, rue Joliot-Curie F-13453 Marseille Cedex 13, France e-mail: [email protected]

Inverse Scattering Problem and Canonical Systems F. Peherstorfer Institute for Analysis Johannes Kepler University Linz A-4040 Linz, Austria e-mail: [email protected] A. Volberg Department of Mathematics Michigan State University East Lansing MI 48824, USA e-mail: [email protected] P. Yuditskii Department of Mathematics Bar Ilan University Ramat Gan 52900, Israel e-mail: [email protected]

323


Remark on the Compensation of Singularities in Krein’s Formula A.B. Mikhailova and B.S. Pavlov Abstract. We reduce the spectral problem for an additively perturbed selfadjoint operator HV = H0 − V , to the dual problem of finding zeros of the operator function Sign V − |V |1/2 [H0 − λ]−1 |V |1/2 , and develop the Schmidt perturbation procedure for the resolvent of HV . Based on Rouche theorem for operator-valued analytic functions, we observe, in the Krein’s formula for the perturbed resolvent [HV − λI]−1 , the compensation of singularities inherited from H0 , and suggest a convenient algorithm for approximate calculation of the groups of eigenfunctions and eigenvalues of the perturbed operator. Mathematics Subject Classification (2000). Primary 47N50; Secondary 35Q40. Keywords. Lippmann-Schwinger equations, eigenvalues, eigenfunctions.

1. Introduction: formal solution of the Lippmann-Schwinger equation and Krein’s formula The non-homogeneous equation with self-adjoint operators H0 , and V in a Hilbert space E (1.1) [H0 − V − λI] u = f can be conveniently reduced, see [1], to the corresponding Lippmann-Schwinger equation −1 −1 (1.2) u = [H0 − λI] f + [H0 − λI] V u := Rλ0 f + Rλ0 V u. The formal solution of this equation leads to the dual problem of localizing zeros of a relevant operator function. Indeed, assuming that the perturbation V = V + is bounded, we represent it as V = |V |1/2 ΘV |V |1/2 with a unitary self-adjoint The authors acknowledge support from the Russian Academy of Sciences, Grant RFBR 03-0100090.

326

A.B. Mikhailova and B.S. Pavlov

operator ΘV = [ΘV ]+ = Sign V acting in EV := |V |1/2 E. Then we rewrite (1.2) as |V |1/2 u = |V |1/2 Rλ0 f + |V |1/2 Rλ0 |V |1/2 ΘV |V |1/2 u, (1.3)

and find |V |1/2 u as

,−1 + |V |1/2 Rλ0 f, |V |1/2 u = IV − |V |1/2 Rλ0 |V |1/2 ΘV

(1.4)

where IV is the restriction of identity to EV . This gives a convenient formula for the solution of (1.2) $ # IV 1/2 0 (1.5) |V | R u = Rλ0 + Rλ0 |V |1/2 λ f. ΘV − |V |1/2 Rλ0 |V |1/2

A similar formula in operator extension theory was first suggested by M. Krein, see [2], and, in different form, by M. Naimark, [3], as a device for the parametrization (of resolvents) of self-adjoint extensions of a given symmetric operator. Later, the idea of reducing the eigenvalue problem for the additively perturbed operator HV = H0 − V to the dual problem of zeros of the denominator D = ΘV − |V |1/2 Rλ0 |V |1/2 appeared in [4, 5], and, since publication [7], is called the Birman-Schwinger principle. All these formulae are based on the reduction of the inversion of an infinite-dimensional operator to the simpler problem of inversion of the corresponding denominator D with a compact or finite-dimensional |V |1/2 Rλ0 |V |1/2 . In this paper, for historical reasons and for sake of brevity, we refer to the corresponding formula for the resolvent −1

RλV := [H0 − V − λI]

= Rλ0 + Rλ0 |V |1/2

IV |V |1/2 Rλ0 ΘV − |V |1/2 Rλ0 |V |1/2

(1.6)

as general Krein’s formula. In the original Krein formula [2] the denominator D is finite-dimensional. The above derivation of the formula, in case of Schr¨ odinger operators with general bounded self-adjoint perturbations V , which are in a certain way subordinated to H0 can be found in [6]. Both summands in the right side of (1.6) are singular at the eigenvalues of the non-perturbed operator H0 . For one-dimensional V , the singularities of the first and second summands in the right side of (1.6) inherited from the unperturbed operator H0 , compensate each other, so that only zeros of the denominator D arise as isolated eigenvalues of the perturbed operator H0 − V . The compensation of the singularity inherited from a single simple eigenvalue of the unperturbed problem was noticed in [8, 9, 10, 11] and used for approximate calculation of resonances in [12]. Though it was commonly expected that the singularities of both summands in the original Krein formula compensate each other in case of isolated singularities, but the proof of the fact for a group of isolated eigenvalues requires a deeper insight into the problem and revealing connections with the Schmidt perturbation procedure, see [1]. The standard analytic perturbation procedure, see [13], is applicable to the problem of perturbation of an isolated eigenvalue λ0 of the operator H0 , if the

Remark on the Compensation of Singularities in Krein’s Formula

327

perturbation V : H0 → HV = H0 − V is dominated by the spacing of the nonperturbed operator H0 at the eigenvalue λ0 : 2 V < min |λs − λ0 | =: ρ(λ0 ). λs =λ0

(1.7)

A modified (“Schmidt”) analytic perturbation procedure sketched in [1] is based on the conjoint perturbation of the eigenvalue λ0 and the nearest to it eigenvalue λ1 to the end of augmentation of the spacing ρ(λ0 ). Our version of the Schmidt procedure permits to extend the tight bounds (1.7) of application of the standard analytic perturbation procedure to the individual eigenvalues by considering the simultaneous perturbation of a group of eigenvalues, situated on a certain “essential” spectral interval ∆ as zeros of the corresponding denominator D, via comparison of it with the relevant model denominator D, based on the operator valued Rouche theorem, [16]. In this paper, based on the announced compensation result for a group of eigenvalues, see Section 2, we obtain in Section 3 convenient formulae for the eigenfunctions and eigenvalues of the perturbed operator. In last section of this paper, based on the compensation of singularities, we reveal a transformation of the intersections of the dispersion curves of a periodic Sturm-Liouvilles problem into quasi-intersections.

2. Compensation of singularities The original Krein formula permits to reduce the calculation of the resolvent of the perturbed operator to the construction of the inverse of the Birman-Schwinger denominator: (2.1) ΘV − |V |1/2 Rλ0 |V |1/2 := D. In case when the condition |V |1/2 Rλ0 |V |1/2 < 1 is fulfilled on a real spectral interval ∆, the inverse D−1 can be calculated based on standard analytic perturbation procedure, see for instance [1], resulting in a corresponding Neumann series. We extend the “Schmidt method” suggested by R. Newton, see [1], to treat the perturbation problem for a group of few eigenvalues of H0 on ∆: σ(H0 ) ∩ ∆ =: σ Δ = {λ1 , λ2 , λ3 , . . . , λN }. We denote by ϕ1 , ϕ2 , ϕ3 , . . . , ϕN the corN responding eigenfunctions, by EΔ the linear hull of {ϕs }s=1 , by P Δ the orthogonal N projection on it in E, and by H Δ = s=1 λs ϕs ϕs the part of H0 in E Δ . Assume, that the perturbation V is subordinated to H0 so that |V |1/2 Rλ0 |V |1/2 is compact on the complement of σ Δ in the complex neighborhood ΩΔ of ∆, see in [14, 15] for properties of compact-valued analytic operator functions. Then the denominator D is a meromorphic function on ΩΔ with a finite number of real poles λs and a finite number of real vector zeros λVs , es : D(λVs ) es = 0, es ∈ E. These zeros are eigenvalues, in ΩΔ , of the perturbed operator, ∪s λVs = σVΔ . Hereafter in this paper we assume that the sets σ Δ , σVΔ , are disjoint, σ Δ ∩ σVΔ = ∅. Following [16] the index of the vector zero zs of D denotes the total dimension of the corresponding root space. For instance, for an elementary Blaschke factor

328


B of degree k in the upper half-plane ℑλ > 0, with a complex zero zs and an orthogonal projection Ps onto the null- space {ν : B(zs )ν = 0} , B(λ) =

λ − zs λ − z¯s

k

Ps + Ps⊥ , Ps + Ps⊥ = I,

the index is equal dim Ps × k. For poles the index is defined similarly, but with negative k. The total index of a meromorphic function in the domain Ω is the (algebraic) sum of all indices at zeros and poles of the function situated in Ω. The operator version of Rouche’s theorem, proved by Gohberg and Sigal, see [16], connects the total indices of two meromorphic operator functions D, D0 in a domain Ω, if the difference D − D0 := δD is relatively small on the boundary: If on the boundary ∂Ω the inequality −1 δD < 1 sup I − D−1 0 D = sup D0 ∂Ω

(2.2)

∂Ω

is fulfilled, then the total indices of D, D0 in Ω coincide. In particular, if the poles of D, D0 coincide and have the same indices, then the total index of zeros of D in Ω coincides with the total index of zeros of D0 in Ω. Combining the idea of Schmidt method, see [1], (2, Chapter 9) with Rouche’s theorem for matrix valued functions, [16], we develop a perturbation procedure and localize the isolated eigenvalues λVs of the perturbed operator. This provides a basement for the proof of the compensation of singularities and the derivation, in the next section, a formula for the eigenfunctions of the perturbed operator. To formulate the main result of this section, we need specific notations. Denote E ⊖ E Δ =: E ⊥ and introduce PE ⊥ =: P ⊥ = I − P Δ . The identity operator in E Δ will be denoted by I Δ = P Δ . We denote by H Δ the part of H0 in Δ E

E Δ and by H ⊥ the part of H0 in E ⊥ , the spectrum of H ⊥ will be denoted by σ ⊥ . The resolvents of H Δ and H ⊥ will be denoted by RλΔ and Rλ⊥ respectively, s . A small complex neighborhood of ∆, specified below, will be RλΔ = λs ∈Δ λsP−λ denoted by ΩΔ . In our analysis, when using the operator-valued Rouche theorem, we choose D0 =: Dlead := ΘV − |V |1/2 RλΔ |V |1/2 and substitute the condition (1.7) by the condition imposed onto δD = D⊥ = |V |1/2 Rλ⊥ |V |1/2 , on a selected neighborhood ΩΔ , such that sup D⊥ (λ) = sup |V |1/2 Rλ⊥ |V |1/2 < 1 − ε, ε > 0.

λ∈ΩΔ

λ∈ΩΔ

Due to the maximum principle the weaker condition sup D⊥ (λ) = sup |V |1/2 Rλ⊥ |V |1/2 < 1 − ε, ε > 0,

λ∈∂ΩΔ

λ∈∂ΩΔ

(2.3)


329

is equivalent to (2.3). On the other hand, the following slightly stronger condition also implies (2.3): sup |V |1/2 P ⊥ |V |1/2 ≤ (1 − ε)

λ∈ΩΔ

inf

λ∈ΩΔ , λs ∈C\ΩΔ

|λ − λs | = (1 − ε)dist(λ, σ ⊥ ). (2.4)

We consider the matrix function acting in EV D = ΘV − |V |1/2 Rλ⊥ |V |1/2 = ΘV − D⊥ .

(2.5)

The matrix D is invertible on ΩΔ since (2.3) due to the Banach principle, because of the geometrical convergence of the series: D−1 = ΘV + ΘD⊥ Θ + ΘD⊥ ΘD⊥ Θ + · · · . Lemma 2.1. If (2.3) is fulfilled on ΩΔ , then D is invertible and P Δ |V |1/2 D−1 (λ)|V |1/2 P Δ =: V (λ)

is analytic on ΩΔ . It is represented by the geometrically convergent series P Δ |V |1/2 D−1 (λ)|V |1/2 P Δ = P Δ V P Δ + P Δ |V |1/2 [D−1 − ΘV ]|V |1/2 P Δ

=: P Δ V P Δ + P Δ V Rλ⊥ V P Δ + P Δ V Rλ⊥ V Rλ⊥ V P Δ + · · · =: P Δ V P Δ + ΣΔ ,

where ΣΔ (λ) = P Δ |V |1/2 Moreover

l 1−ε . ΘV D⊥ ΘV |V |1/2 P Δ ≤ P Δ |V |1/2 2 ε l≥1

dΣΔ > 0. dλ Proof. The proof of the statements can be obtained via a straightforward estimation of the above series. For instance (2.3) implies the above estimation for ΣΔ on ΩΔ and l ΣΔ = P Δ |V |1/2 ΘV D⊥ ΘV |V |1/2 P Δ l≥1

hence

= P Δ V [Rλ⊥ + Rλ⊥ V Rλ⊥ + · · · ]V P Δ I V P Δ, = P ΔV ⊥ H0 − λI ⊥ − P ⊥ V P ⊥

dΣΔ I Δ = P ΔV 2 V P > 0. dλ H0⊥ − λI ⊥ − P ⊥ V P ⊥

We will also use the analytic matrix function d in E Δ

d = H Δ − λI Δ − P Δ |V |1/2 D−1 |V |1/2 P Δ =: H Δ − λI Δ − V (λ) = H Δ − λI Δ − P Δ V P Δ − ΣΔ (λ).

330


Theorem 2.1. If, for given V = V + , there exist a domain ΩΔ such that the above condition (2.3) is fulfilled, then the total multiplicity of eigenvalues of the perturbed operator H0 − V in ΩΔ is equal to the total multiplicity of the eigenvalues of H0 on ∆ and the perturbed eigenvalues can be found as zeros of d. The resolvent of the perturbed operator is represented on ΩΔ by the modified Krein formula RλV = Rλ⊥ + Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 P ⊥ Rλ⊥ (2.6) , + , + Δ Δ 1/2 −1 1/2 ⊥ ⊥ Δ ⊥ ⊥ 1/2 −1 1/2 Δ I P + P |V | D |V | P Rλ . + P + Rλ P |V | D |V | P d

Proof. The group of leading terms of the denominator on the spectral interval ∆ includes ΘV and the part of the spectral expansion of the unperturbed resolvent in EΔ : P Δ Rλ0 P Δ = RλΔ , squeezed between the factors |V |1/2 . Dlead := ΘV − |V |1/2 RλΔ |V |1/2 = ΘV +

N |V |1/2 ϕs |V |1/2 ϕs s=1

λ − λs

.

(2.7)

The rest D⊥ (λ) := |V |1/2 Rλ⊥ |V |1/2 of D, containing the part of the unperturbed resolvent Rλ⊥ in the complementary subspace, E ⊥ = E ⊖ E Δ can be estimated on ΩΔ , due to (2.4), as D⊥ (λ) ≤

|V |1/2 P ⊥ |V |1/2 ≤ 1 − ε. dist(σ ⊥ , λ)

(2.8)

Due to the above spectral estimate of D⊥ and (2.3) D is invertible for λ ∈ ΩΔ . Note that functions D = Dlead + D⊥ and Dlead have the same singularities in ΩΔ . Then, based on the quoted operator-valued version of Rouche theorem, one can localize the zeros of D comparing D with Dlead . The inverse of D is calculated on ΩΔ via Neumann series. We calculate the inverse of D, solving the equation: (2.9) D − |V |1/2 RλΔ |V |1/2 u = |V |1/2 Rλ0 f, f ∈ E. Multiplying it by D−1 and introducing the new variable v=

N s=1

vs ϕs = RλΔ |V |1/2 u,

with vs = (λs − λ)−1 |V |1/2 ϕs , u , we consider the dot-product of the result with |V |1/2 ϕt : (λt − λ)vt −

N s=1

|V |1/2 ϕt , D−1 |V |1/2 ϕs vs = |V |1/2 ϕt D−1 |V |1/2 Rλ0 f .

(2.10)

This equation for v can be reduced, via multiplication (2.10) by ϕt , and subsequent summation over t, to Δ H − λI Δ v − P Δ |V |1/2 D−1 |V |1/2 P Δ v = P Δ |V |1/2 D−1 |V |1/2 Rλ0 f. (2.11)


331

The analytic matrix function H Δ − λI Δ − P Δ |V |1/2 D−1 |V |1/2 P Δ = d is invertible for all λ ∈ ΩΔ , except the zeros of det d. Then, on the complement of the set of zeros we have: v = d−1 P Δ |V |1/2 D−1 |V |1/2 Rλ0 f. hence (2.11) implies:

u = D−1 |V |1/2 v + D−1 |V |1/2 Rλ0 f + , = D−1 |V |1/2 P Δ d−1 P Δ |V |1/2 D−1 |V |1/2 + I Rλ0 f.

Inserting the above result into the above Krein formula (1.6), we obtain + , RλV = Rλ0 + Rλ0 |V |1/2 D−1 |V |1/2 P Δ d−1 P Δ |V |1/2 D−1 |V |1/2 + I Rλ0 .

(2.12)

(2.13)

Both summands in the right side of (2.13) have singularities at the spectrum of H0 . But we are able to observe compensation of the singularities. Let us represent the left and right resolvent factors Rλ0 in the right side of (2.13) as Rλ0 = P Δ Rλ0 + P ⊥ Rλ0 , and consider the matrix elements of the second term in the right side of (2.13), Δ ⊥ with respect to the decomposition E = E ⊕ E , denoting the square bracket in (2.13) as P Δ d−1 P Δ |V |1/2 D−1 |V |1/2 + I = [∗]: RλΔ P Δ |V |1/2 D−1 |V |1/2 [∗]P Δ RλΔ =: Δ {∗} Δ , Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 [∗]P Δ RλΔ =: ⊥ {∗} Δ ,

RλΔ P Δ |V |1/2 D−1 |V |1/2 [∗]P ⊥ Rλ⊥ =: Δ {∗} ⊥ , Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 [∗]P ⊥ Rλ⊥ =: ⊥ {∗} ⊥ .

The square bracket in the first expression can be substituted by [∗]P Δ = P Δ d−1 P Δ |V |1/2 D−1 |V |1/2 P Δ + P Δ = d−1 [H Δ − λI Δ ]. Then adding to obtain RΔ +

Δ

Δ

{∗} Δ the component RλΔ of the first summand Rλ0 in EΔ , we

+ , {∗} Δ = RλΔ I + P Δ |V |1/2 D−1 |V |1/2 P Δ d−1 P Δ

= RλΔ [H0Δ − λI Δ ] P Δ d−1 P Δ = P Δ d−1 P Δ ,

thus, the singularities of first expression at the eigenvalues of by the singularities of the first term RλΔ . The second expression ⊥ {∗} Δ is represented as ⊥

H0Δ

(2.14)

are compensated

{∗} Δ = Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 [∗]P Δ RλΔ

= Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 d−1 [H Δ − λI Δ ] RλΔ

= Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 P Δ d−1 P Δ .

(2.15)

332


Hence it is not singular at the eigenvalues of H0Δ . Similarly the third expression is transformed: Δ −1 d |V |1/2 D−1 |V |1/2 Rλ⊥ . Δ {∗} ⊥ = P

The last expression on ∆: ⊥

⊥

{∗} ⊥ does not have singularities at the eigenvalues of H0

{∗} ⊥ = Rλ⊥ |V |1/2 D−1 |V |1/2 P Δ d−1 P Δ |V |1/2 D−1 |V |1/2 Rλ⊥ − Rλ⊥ |V |1/2 D−1 |V |1/2 Rλ⊥ .

(2.16)

Summarizing the results derived, we obtain the announced form of the Krein formula in the domain Ωλ : RλV = Rλ⊥ + Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 P ⊥ Rλ⊥ , + , + + P Δ + Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 P Δ d−1 P Δ + P Δ |V |1/2 D−1 |V |1/2 P ⊥ Rλ⊥

The only singularities in the right side of the above expression in ΩΔ are the zeros λVt ∈ ∆ of the N × N analytic matrix function d in agreement with the elementary spectral perturbation result that the shift of the spectrum of H0 does not exceed the norm of the perturbation. These zeros of d define the eigenvalues of the perturbed operator H0 − V in ΩΔ . Really, the left and right factors P ⊥ + Rλ⊥ P ⊥ |V |1/2 D−1 |V |1/2 P Δ , P ⊥ + P Δ |V |1/2 D−1 |V |1/2 P ⊥ Rλ⊥

(2.17)

are invertible analytic functions on ∆, hence none of the vector zeros of d on ∆ can be compensated by the corresponding zero of the factor.

3. Spectral results Corollary 3.1. Group localization of eigenvalues. Assume that, together with condition (2.3), the group localization of zeros of the operator function d(λ) = H Δ − λI Δ − P Δ V (λ)P Δ = H Δ − λI Δ − P Δ V P Δ − ΣΔ (λ) is observed: 1. The discs Bρs centered at the eigenvalues λVs of the operator H Δ −P Δ V P Δ , with radii ρ, P Δ |V |1/2 ΘV P ⊥ 2 ρ> (3.1) ε are contained in ΩΔ . 2. The union ∪s Bρs = B is represented as a sum of non-overlapping connected components Bl : B = ∪l Bl . Then the total multiplicity of eigenvalues of the perturbed operator H0 − V in each component Bl coincides with the multiplicity of eigenvalues of HVΔ := H Δ − P Δ V P Δ in Bl .


333

Consider, together with the denominator d, the “shortened” denominator H Δ − λI Δ − P Δ |V |1/2 ΘV |V |1/2 P Δ =: H Δ − λI Δ − P Δ V P Δ =: d1 (λ).

Estimate the deviation of the ratio [d1 (λ)]−1 d(λ) from the unity. We can estimate the difference d(λ) − d1 (λ) = ΣΔ based on Lemma 2.1. Then, due to the spectral estimation of the resolvent of HVΔ −1 HVΔ − λI Δ ≤ ρ−1 , we obtain:

1−ε ≤ 1 − ε. ρε Hence the indices of zeros of d and d1 in each connected component Bl are equal due to the matrix Rouche theorem. Since both functions are analytic in ΩΔ , the total multiplicity of zeros of d on each connected component Bl is equal to the total multiplicity of zeros of d1 . Choosing ρ, ε properly, we may assume that the first condition Bρs ∈ ΩΔ is always fulfilled. Δ 1/2 2 I − d(λ)d−1 1 (λ) ≤ P |V |

Corollary 3.2. The perturbed spectral projections. The spectral projections at the isolated eigenvalues of a self-adjoint operator coincide with the residues of the corresponding polar terms in the resolvent. Though the operator function d = H Δ − λI Δ − P Δ V P Δ − Σ(Λ) is self-adjoint if λ ∈ ∆, we should prove that all poles of d−1 are simple. Note first that all dΣ δ V zeros of d are simple, because dd dλ = I + dλ ≥ I. Now, assuming that λ0 , e0 is a simple vector zero of d, and denoting e0 e0 =: P0 , we represent d in close vicinity of λ0 as dΣΔ + O (λ − λV0 )2 . dλ Neglecting the quadratic error, we represent, in the vicinity of λ0 , the equation d(λ) = H Δ − λI Δ − P Δ V (λ0 )P Δ − (λ − λV0 )

d(λ)u = f −1 Δ Δ Δ = Rλ0 as via the resolvent H − λI − P V (λV0 )P Δ u = Rλ0 f − (λ − λV0 )Rλ0

dΣΔ V dΣΔ (λ0 )u ≈ Rλ0 f − P0 u, dλ dλ

with the projector P0 = e0 e0 . Due to Lemma 2.1 e0 , a small vicinity of λV0

dΣΔ V dλ (λ0

)e0 > 0, hence in

dΣΔ V 0 (λ0 )Rλ f P0 dλ )e0 1 P f + ··· . ≈ Δ V )e λV − λ 0 (λ 1 + e0 , dΣ 0 0 0 dλ

u = Rλ0 f −

1

1 + e0 , 1

dΣΔ V dλ (λ0

334


Denoting by J, J + the left and the right factors (2.17), we calculate the polar term of the perturbed resolvent at λ0 1 1 + e0 ,

dΣΔ V dλ (λ0

JP0 J + . )e0 λV0 − λ

Then we conclude from the above formula that generally, for the simple vector zeros of d, index one, the following holds The vector zeros λVs , νsV of d are simple eigenvalues of the perturbed operator H0 − V and the corresponding spectral projections are PsV =

1+

νsV

JP0 J + Δ V V , dΣ dλ (λs )νs

The eigenfunctions of the perturbed operator may be obtained from the eigenfunc tions νsV of the operator H Δ − P Δ V (λVs )P Δ V as: Ψs = JνsV ,

with subsequent normalization.

+ In case of multiple zeros of d the factor 1 + νsV , tuted by the corresponding operator-valued function.

dΣΔ V dλ (λs

,−1 is substi)νsV

4. Example While the standard analytic perturbation technique is aimed on a description of the behavior of a single eigenvalue under a small perturbation, our version (2.6) of Krein’s formula permits to observe the behavior of groups of eigenvalues under perturbations. Note that considering groups of eigenvalues we do not connect the eigenpairs of the perturbed and the ones of the unperturbed operator “one-toone” (individually), but just establish the correspondence between groups. This additional freedom permits to observe transformation of the intersection of terms into the quasi-intersection and details of the Landau-Zener phenomenon, see for instance [18, 19]. In this section we consider the behavior of the eigenvalues and eigenfunctions of the quasi-periodic Sturm-Liouvilles problem on a finite interval under a perturbation defined by a real continuous potential V . The role of the unperturbed operator in L2 (0, 1) is played by H0 (p)u = −u′′ , u ∈ W22 (0, 1), u(1) = eip u(0), u′ (1) = eip u′ (0)

(4.1)

ψ0l (x, p) = ei(2lπ+p)x , λl0 (p) = (2lπ + p)2 .

(4.2)

with quasi-momentum p, 0 < p < 2π. The unperturbed eigenpairs are The perturbed operator, with the potential V is defined as HV (p)u = −u′′ + V u, u ∈ W22 (0, 1),

u(1) = eip u(0), u′ (1) = eip u′ (0), 0 < p < 2π.

(4.3)


335

It also has discrete spectrum with corresponding eigenpairs ψVl (x, p), λlV (p).

(4.4)

The asymptotic behavior of the eigenvalues λlV (p) = λl0 (p) + O(1)

(4.5)

with uniformly bounded error O(1) and the corresponding asymptotic of eigenfunctions is obtained by WKB methods, see for instance [17]. The low-energy results are obtained for a small potential by the analytic perturbation technique, see for instance [20]. The above Krein formula (2.6) permits to describe the low-energy spectral properties of the quasi-periodic operator for any real continuous potential. Note, first of all, that the spacing between consecutive eigenvalues λl0 (p), (p) of the unperturbed operator is growing linearly with l. Hence one always λl+1 0 can select, for given continuous real potential V , a spectral interval ∆, such that dist (σ Δ (p), σ ⊥ (p)) > 2 max |V |. Moreover, one can select ∆ such that the parameter ε in the condition (3.1) is large enough, to guarantee that the radius ρ estimated in (3.1) is as small, as we need. Then the search of eigenvalues of the perturbed operator, reduced by Theorem 2.1 to the search of zeros of the corresponding function d, is substituted by the much easier search of eigenvalues and eigenfunctions of the finite matrix H0 − P Δ V P (∆), with an error ρ. For the calculation of eigenvalues and eigenfunctions of the classical Mathieu equation with V (x) = cos 2πx 2 ⊥ for π/2 > 1 = max |V (x)|, so that 4 2 ε = 15π 4 − 1 > 32. Then ρ can be chosen as 0.03. The original periodic spectral problem in L2 (R) is substituted by the spectral problem with 4 × 4 matrix: ⎛ −1 ⎞ λ0 0 0 0 0 ⎜ 0 λ0 0 0 ⎟ ⎟ d1 (λ, p) = H0 − P Δ V P Δ = ⎜ ⎝ 0 0 λ10 0 ⎠ 0 0 0 λ20 ⎛ ⎞ −1 −1 −1 −1 0 ψ0 , V ψ0 ψ0 , V ψ0 ψ0 , V ψ01 ψ0−1 , V ψ02 −1 2 0 1 0 0 0 0 ⎜ ψ0 , V ψ ψ0 , V ψ0 ⎟ ψ0 , V ψ0 ψ0 , V ψ0 0 ⎟. +⎜ ⎝ ψ01 , V ψ −1 ψ01 , V ψ00 ψ01 , V ψ01 ψ01 , V ψ02 ⎠ 0 −1 2 2 0 2 1 2 2 ψ0 , V ψ0 ψ0 , V ψ0 ψ0 , V ψ0 ψ0 , V ψ0

0 1 2 Here λ−1 0 , λ0 , λ0 , λ0 , are the eigenvalues of the unperturbed operator, with zero −1 potential, and ψ0 , ψ00 , ψ01 , ψ02 are the corresponding eigenfunctions. The group of unperturbed eigenvalues is situated on the interval [0, (7π/2)2 ] and reveal an intersection of terms depending in quasi-momentum p. The perturbed terms λlV , l = −1, 0, 1, 2, with non-trivial V = 0, as functions of p reveal typical quasiintersections. The first order approximation of the perturbed therm is found as zeros of the determinant det d. More precise results can be obtained in course of analysis of the corresponding 4 × 4 matrix H0 − P Δ |V |1/2 ΘV |V |1/2 P Δ .

336


Acknowledgement The authors are grateful to V. Kruglov and P. Kurasov for important references, and to the referee for useful remarks and suggestions.

References [1] R. Newton, Scattering theory of waves and particles Dover Publications, Inc. Mineola, NY, 2002, 745 p. [2] M. Krein, Concerning the resolvents of an Hermitian operator with deficiency index (m, m), Doklady AN USSR ,52, 651 (1946). [3] M. Naimark, Self-adjoint extensions of the second kind of a symmetric operators, Bull. AN USSR, Ser. Math. 4, 53 (1946). [4] J. Schwinger, On bound states of a given potential Proc. Nat. Acad. Sci. USA, 47 (1961) pp. 122–129. [5] M.Sh. Birman, On the spectrum of singular boundary-value problems Mat. Sbornik (N.S.) 55 (1961) pp. 125–174. English transl. in Amer. Math. Soc. Transl. 53 (1966), pp. 23–80. [6] D.R. Yafaev, Mathematical scattering theory. General Theory Amer. Math. Soc., Providence, RI, (1992) 341 p. [7] B. Simon, Trace Ideals and Their Applications London Math. Soc. Lecture Notes Series, 35. Cambridge Univ. Press, Cambridge, NY (1979) 134 p. [8] V. Bogevolnov, A. Mikhailova, B. Pavlov, A. Yafyasov, About Scattering on the Ring In: “Operator Theory: Advances and Applications”, Vol. 124 (Israel Gohberg Anniversary Volume), Ed. A. Dijksma, A.M. Kaashoek, A.C.M. Ran, Birkhäuser, Basel (2001) pp. 155–187. [9] A. Mikhailova, B. Pavlov, Resonance Quantum Switch In: S. Albeverio, N. Elander, W.N. Everitt and P. Kurasov (eds.), Operator Methods in Ordinary and Partial Differential Equations, (S. Kovalevski Symposium, Univ. of Stockholm, June 2000), Birkh¨ auser, Basel-Boston-Berlin, 2002 (Operator Methods: Advances and Applications 132). [10] A. Mikhailova, B. Pavlov, Quantum Domain as a triadic Relay In: Unconventional models of Computations UMC’2K, ed. I. Antoniou, C. Calude, M.J. Dinneeen, Springer Verlag series for Discrete Mathematics and Theoretical Computer Science (2001) pp. 167–186. [11] A. Mikhailova, B. Pavlov, L. Prokhorov Modelling of Quantum Networks arXiv mathph/031238, 2004, 69 p. [12] N. Bagraev, A. Mikhailova, B. Pavlov, L. Prokhorov, A. Yafyasov, Parameter regime of a resonance quantum switch, Phys. Rev. B, 165308 (2005). [13] T. Kato, Perturbation theory for linear operators Springer Verlag, Berlin-HeidelbergNY, second edition (1976). [14] Y.P. Ginzburg, Almost invariant properties of analytic operator families (Russian) Funktsional. Anal. i Prilozhen. 16 (1982), no. 1, 68–69.


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[15] Y.P. Ginzburg, L.M. Zemskov, Multiplicative representations of operator-functions of bounded type (Russian) Teor. Funktsi˘ı Funktsional. Anal. i Prilozhen. No. 53 (1990), 108–119; translation in J. Soviet Math. 58 (1992), no. 6, 569–576. [16] I.S. Gohberg and E.I. Sigal, Operator extension of the theorem about logarithmic residue and Rouch´ e theorem Mat. Sbornik. 84, 607 (1971). [17] E.C. Titchmarsh, Eigenfunctions expansions associated with second order differential equations, Clarendon Press, Oxford, 1 (1962). [18] C. Zener, Non-adiabatic crossing of energy-levels Proc. R. Soc. A, 137, (1932) 696. [19] Y.N. Demkov, P.B. Kurasov, V.N. Ostrovski, Double-periodical in time and energy solvable system with two interacting set of states Journal of Physics A, Math. and General, 28, (1995), 434. [20] V. Arnold, Remarks on perturbation theory for problems of Mathieu type, Russian Math. surveys 38, 4, (1983), 215–233. A.B. Mikhailova V.A. Fock Institute for Physics St. Petersburg State University Ulyanovskaya 1, Petrodvorets St. Petersburg, 198504, Russia B.S. Pavlov Department of Mathematics The University of Auckland Private Bag 92019 Auckland, New Zealand e-mail: [email protected]


On Some Properties of Infinite-dimensional Elliptic Coordinates Andrey Osipov Abstract. A number of the equations in classical mechanics is integrable in Jacobi elliptic coordinates. In recent years a generalization of elliptic coordinates to the infinite case has been offered. We consider this generalization and indicate the connection of infinite-dimensional elliptic coordinates with some inverse spectral problems for infinite Jacobi matrices and Sturm-Liouville operators (two spectra inverse problems). Also a link with the rank one perturbation theory is shown. Mathematics Subject Classification (2000). 47B36; 49N45, 81Q10. Keywords. Elliptic coordinates, Inverse spectral problems, Weyl function, Herglotz function, Rank one perturbation.

1. Introduction One of the most powerful integration methods for equations of motion in classical mechanics is separation of variables. In a number of important cases there exists a substitution known as Jacobi elliptic coordinates which allows one to find such separating variables. For example such problems of classical mechanics as plane motion in the field of two attracting centers, Kepler’s problem in the homogeneous force field and geodesic motion on an n-dimensional ellipsoid are integrable in elliptic coordinates [1]. They can be introduced in the following manner. Consider the space H = Cp (or Rp ), a symmetric operator A in H with simple eigenvalues and a vector x ∈ H. Then we can define the elliptic coordinates (e. c.) of the vector x as the set of numbers {γn }, which for some fixed C > 0 satisfies the equation (Rλ x, x) = C,

Rλ = (A − λE)−1 .

This work has been supported by the Russian Foundation For Basic Research: project 05-0100989 and by the Ministry for Science and Technology of Russia: project NSh-5247.2006.1 .

340

A. Osipov

If we take the orthonormal basis {en }pn=1 in H such that Aen = λn en , and λn < λn+1 , n = 1, . . . , p − 1 then the above equation can be written as p |xn |2 = C, λ −λ n=1 n

xn = (x, en ).

Assume that xn = 0 for all n. One can easily check that the above equation has exactly p roots and γ1 < λ1 , λn−1 < γn < λn , n = 2, . . . , p. For the coordinates xi we have (see, e.g., [1]): p (λi − γn ) |xi |2 = C n=1 , i = 1, . . . , p. (γ n =i i − γn )

V.I. Arnold posed the following question [1]–[2]: Generalize Jacobi elliptic coordinates to the infinite-dimensional setting and find equations of mathematical physics integrable by this method. The first step in this direction has been made by Kostyuchenko and Stepanov in [3]. Considering a separable Hilbert space H instead of Cp and applying the above operator approach they found the necessary and sufficient conditions for a sequence {γn } to be the e. c. of a certain vector x ∈ H. Unfortunately these conditions are too restrictive for some cases important for the integration task. In [4] Vaninsky considered a more general concept of e. c. which is operator-free. He has shown that the Camassa-Holm equation (approximation to the Euler equation describing an ideal fluid) is integrable in e. c. introduced in this manner. The aim of this paper is to indicate the connection between both concepts of e. c. and some two-spectra inverse problems. It allows one to formulate some known results on these inverse problems in terms of e. c. Also our examples show that both concepts of elliptic coordinates can be embedded into the rank one perturbation framework. The first concept corresponds to the bounded perturbations and the second concept to the singular ones.

2. First definition of infinite e. c. and link to inverse problems Consider a self-adjoint lower semibounded operator A in a separable Hilbert space H with a simple purely discrete spectrum {λn }∞ n=1 , λ1 < λ2 < · · · < λn → ∞ as n → ∞. Let {en }∞ n=1 be its orthonormal eigenbasis. As in [3], define the elliptic coordinates {γn }∞ n=1 of a certain cyclic vector x ∈ H as the ordered roots of the equation (1). If we define a function ∞ |xn |2 − C, xn = (x, en ), F (λ) = λ −λ n=1 n

then (1) is equivalent to the equation F (λ) = 0. This seems to be the most natural way to generalize the above definition of elliptic coordinates to the infinitedimensional case. F (λ) is a Herglotz function, so its zeros and poles interlace: γ1 < λ1 < γ2 < λ2 < · · · .

(2.1)

Infinite-dimensional Elliptic Coordinates

341

Theorem (Kostyuchenko-Stepanov). Let the operator A is defined as above. A sequence {γn }∞ n=1 satisfying (2.1) represents the elliptic coordinates of some vector x ∈ H and cyclic for A, iff the following condition holds ∞ (λn − γn ) < ∞. (2.2) n=1

In this case for the coordinates xi = (x, ei ) of x the following equations hold: ∞ (λi − γn ) , i∈N (2.3) |xi |2 = C n=1 n =i (γi − γn ) C

∞

n=1

(λn − γn ) =

∞ i=1

|xi |2

(2.4)

It is interesting to note (see [4]), that (2.4) is a direct generalization to the infinite-dimensional case of an ancient formula of Boole. Now we discuss one inverse problem for semi-infinite Jacoby matrices considered in [5]. Namely, consider the matrix ⎛ ⎞ b 1 a1 0 0 ⎜ a 1 b 2 a2 0 ⎟ J =⎝ ⎠ an > 0, bn ∈ R, n ∈ N, .. .. .. . . . 0

such that the limit point case (see, e.g., [6]) holds, and the spectrum of the corresponding self-adjoint operator J in the space l2 (N) is purely discrete with the 2 accumulation point at +∞. Denote by {en }∞ n=1 is the canonical basis in l (N). be the ordered sequence of eigenvalues of J (they are always simple). Let {λn }∞ n=1 Also consider the following bounded rank one perturbation of J: Jh = J + h(e1 , ·)e1 ,

(2.5)

where h is a negative integer. Denote by {µn }∞ n=1 its ordered spectrum (which is also purely discrete). The problem studied in [5] is to find J and h for a given ∞ {λn }∞ n=1 and {µn }n=1 . First of all, we recall the definitions of the basic objects in the theory of Jacobi operators [6]. Namely, for the operator J one

can introduce the right-continuous resolution of identity E(t) such that J = R tdE(t). Then the spectral density function ρ(t) of J is defined as follows: ρ(t) = (E(t)e1 , e1 ).

Vector e1 is cyclic for J. Also consider the function dρ(t) . m(λ) = ((J − λE)−1 e1 , e1 ) = R t−λ which is called the Weyl function of J. The solution f = {fn }∞ n=1 of the equation Jf = λf,

λ∈C

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is uniquely determined if one gives f1 = 1. For the elements of this solution the following notation is standard: Pk (λ) = fk+1 , k ∈ Z+ , where the polynomial Pk (of degree k) is referred to as the kth orthogonal polynomial of the first kind associated ∞ 2 with J. If for some λ = λn , k=0 |Pk (λn )| < ∞, then λn is the eigenvalue of ) is the corresponding eigenvector. The coefficients an operator J and f = f (λ n ∞ αn = k=0 |Pk (λn )|2 are called the normalizing factors. For the considered operator J the corresponding ρ(t) and m(λ) are given by the formulas. ∞ 1 1 . (2.6) , m(λ) = ρ(t) = αn α (λ − λ) n n n=1 Also

λn ≤t

1 = (e1 , e1 ) =

R

dρ(t) =

∞ 1 . α n=1 n

(2.7)

In general, ρ(t) or m(λ) uniquely determines J [6]. In our case it means that the ∞ knowledge of {λn }∞ n=1 and {αn }n=1 is enough for the unique reconstruction of J. It was shown in [5] that the normalizing factors of J can be calculated by the formula −h n =i (µi − µn ) ∞ αi = , i ∈ N, (2.8) λi − µi n=1 (λi − µn ) and the following trace formula holds ∞ (λn − µn ) = −h. (2.9) n=1

Therefore (2.8)–(2.9) give us a key to the solution of the above-stated inverse problem. Also in [5] some sufficient conditions for a pair of sequences to be the spectra of a Jacobi operator and its rank one perturbation were given. On can easily obtain (2.8)–(2.9) by using the Kostyuchenko-Stepanov theorem. Indeed, according to the Aronzajn-Krein formula m(λ) mh (λ) = 1 + hm(λ)

where mh (λ) is the Weyl function of Jh . Therefore {µn } being the poles of mh (λ), satisfy the equation ∞ 1 1 + = 0. α (λ − λ) h n n n=1 For the normalized eigenvectors fˆn of J we also have 1 P0 (λn ) = √ , n ∈ N, (fˆn , e1 ) = √ αn αn

so {µn } are the elliptic coordinates of the vector e1 for C = − h1 > 0. Then the formulas (2.8)–(2.9) follow directly from (2.3)–(2.4) and (2.7). In view of the above we obtain the following result.


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Proposition 2.1. The spectrum of J and the elliptic coordinates {γn }∞ n=1 of e1 for C = − h1 > 0, uniquely determine J and h, and {γn } coincide with the spectrum of Jh defined by (2.5). Now consider another vector φ, cyclic for J. Define J˜ = J + h(φ, ·)φ, h < 0 and Fh (λ) =

R

dh τ (t) , t−τ

where dh τ (t) is the spectral measure for φ associated to J˜h . Then applying the Aronzajn-Krein formula we obtain that the eigenvalues of J˜h satisfy the equation ∞

τn 1 + = 0, λ − λ h n n=1

τn = (fˆn , φ).

therefore they are e. c. of φ. Conversely, if we take the e. c. {γn }∞ n=1 of some φ for C > 0, then one can check that {γn } is the spectrum of J − C1 (φ, ·)φ Note that according to the Stone theorem (see, e.g., [6]) the above-defined operators A and J are equivalent. Thus we are coming to the following conclusion. Theorem 2.2. Let the operator A in H be defined as above. Then for an arbitrary cyclic x ∈ H its elliptic coordinates for any C > 0 and the spectrum of its rank one perturbation 1 A − (x, ·)x C are identical.

3. More general definition and Sturm-Liouville inverse problems As we see, the condition (2.2) plays an important role in the above constructions. However, as pointed in [4], for the string spectral problem associated to the Camassa-Holm equation, (2.2) does not hold for the spectra of the string with different types of boundary conditions. To overcome this obstacle, in ([4]) the infinite elliptic coordinates were introduced in a more general way. Namely, pick some C > 0 and consider the function ∞ n −C (3.1) F (λ) = n=1 λnA−λ An > 0, n ∈ N; λn < λn+1 → ∞ as n → ∞,

and satisfying the condition ∞ An λn 0

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Theorem 3.1. The sequence {γn }∞ n=1 represents the elliptic coordinates arising from the sequence {An }∞ n=1 satisfying (3.2) and some C > 0 iff {γn } interlace {λn }: γn < λn < γn+1 , n ∈ N; λN −1 < 0 < γN +1 , and the following condition holds:

∞ ′ λn 0 and

∞

n=1

An < ∞. 1 + λ2n

˜ = − ∞ ′ λn > −∞ we have A = 0 and Since limλ→−∞ F (λ) n=1 γn ∞ An λn < ∞. 1 + λ2n n=1

Thus F˜ (λ) admits the representation (3.1).

A similar theorem has been proved in [4] (Theorem 5.2). The above theorem is more suitable for our case.


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To see the examples of these e. c. consider the self-adjoint operator Lh = d2 2 − dx 2 + q(x) in L (0, π) with boundary conditions y ′ (0) + hy(0) = 0, y ′ (π) + Hy(π) = 0,

(3.4) (3.5)

where h, H ∈ R and q(x) ∈ CR [0, π]. Denote by {λhn }∞ n=1 its ordered eigenvalues and by φn = φhn the corresponding eigenfunctions such that φn (0) = 1 and φ′n (0) = −h. The following asymptotic formulas are well known.

π λhn = (n − 1)2 + π2 H − h + 21 0 q(τ )d(τ ) + o n1 , (3.6)

π (3.7) αhn = 0 φ2n (x)d(x) = π2 + o n1 . The spectral density function ρh (t) of Lh and its Weyl function mh (λ) are defined similarly to (2.6) (αn = αhn ). Now pick two values of h : h0 = 0 and h1 > 0. Then, as shown in [8], the eigenvalues of Lh1 , being the poles of mh1 (λ), satisfy the following equation m0 (λ) ≡

∞

n=1

1 1 = . α0n (λ0n − λ) h

As we see from (3.6–3.7), for the spectrum of L0 and {γn = λhn1 } the condition (2.2) is no longer valid, whereas the condition (3.3) is valid. This allows us to formulate the following result. Proposition 3.2. For the operator L0 and any h > 0 the spectrum of Lh represents the elliptic coordinates arising from the sequence { α10 } and C = h1 . n

The operators Lh can be considered as singular rank one perturbations of L0 (see, e.g., [9]). Lh = L0 − hφ, · φ, where φ = δ0 , the delta function at zero. So for this case, similarly to (1) we can define the elliptic coordinates of the singular vector φ for some C > 0 as the roots of the equation φ, (L0 − λ)−1 φ = C, and the spectrum of Lh coincide with the e. c. introduced in this manner for h = C1 . Finally note that one may obtain a similar results for the Sturm-Liouville operators considered in L2 (0, ∞), having a discrete spectrum and satisfying (3.4) ([8], [10]).

References [1] V.I. Arnold, Mathematical methods of classical mechanics (Russian) Nauka, 1989. [2] V.I. Arnold, Arnold’s problems. Springer-Verlag, 2004. [3] A.G. Kostyuchenko, A.A. Stepanov, Infinite-dimensional elliptic coordinates. Funct. Anal. Appl. 33 No 4, (1999), 300–303.

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[4] K.L. Vaninsky, Equations of Camassa-Holm type and Jacobi ellipsoidal coordinates. Commun. Pure Appl. Math. 58 No 9, (2005), 1149–1187. [5] R.Z. Khalilova, On one inverse problem (Russian). Izv. Akad. Nauk Az. SSR, Ser. Fiz.-Tekh. Mat. Nauk 3-4 (1967), 169–175. [6] N.I. Akhiezer, The classical problem and some related problems in analysis. Hafner Publishing, N. Y., 965. [7] B.J. Levin, Distribution of zeros of the entire functions. Translations of Mathematical Monographs 5. Amer. Math. Soc., Providence, 1980. [8] B.M. Levitan, M.G. Gasymov, On the determination of a Sturm-Liouville equation by two spectra. Russ. Math. Surv. 19 No. 2, (1964), 1–63. [9] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable models in quantum mechanics. Springer-Verlag, 1988. [10] F. Gesztesy, B. Simon, Rank one perturbations at infinite coupling. J. Funct. Anal. 128 No 1, (1995), 245–252. Andrey Osipov Scientific-Research Institute for System Studies Russian Academy of Sciences (NIISI RAN) Nakhimovskii pr. 36-1 Moscow, 117218, Russia e-mail: [email protected]


Finite Difference Operators with a Finite-Band Spectrum F. Peherstorfer and P. Yuditskii Abstract. We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so-called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the 2d + 1-diagonal matrix; the d-root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface is the symbol of the shift operator. We demonstrate an application of our general construction to the particular covering, which generate almost periodic CMV matrices recently widely discussed. Then we study an important theme: covering of one spectral surface by another one and the related transformations on the set of multidiagonal operators (so-called Renormalization Equations). We prove several new results dealing with Renormalization Equations for periodic Jacobi matrices (polynomial coverings) and for the case of a rational double covering. Mathematics Subject Classification (2000). 47B36, 30F35, 47B32. Keywords. Hardy Spaces, Riemann Surfaces, Spectral theory, Jacobi matrices, iterations, expanding polynomials, CMV matrices.

1. Introduction 1.1. Ergodic finite difference operators and associated Riemann surfaces The standard (three-diagonal) finite-band Jacobi matrices [9, 27] can be defined as almost periodic or even ergodic Jacobi matrices with absolutely continuous spectrum that consists of a finite system of intervals. We wish to find a natural extension of this class of finite difference operators to the multi-diagonal case. Partially supported by the Austrian Founds FWF, project number: P20413–N18 and Marie Curie International Fellowship within the 6th European Community Framework Programme, Contract MIF1-CT-2005-006966.

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The correspondence between periodic difference operators and algebraic curves (spectral surfaces) was discussed in detail in [18]. Let us generalize this construction to show (at least on a speculative level) how almost periodic or ergodic operators give rise to a corresponding spectral surface. Recall the definition of an ergodic operator [7, 19], see also [29]. Let (Ω, A, dχ) be a separable probability space and let T : Ω → Ω be an invertible ergodic transformation, i.e., T is measurable, it preserves dχ, and every measurable T (d) invariant set has measure 0 or 1. Let {q (k) }dk=0 be functions from L∞ dχ , with q (0) positive-valued and q real-valued. Note that in the periodic case Ω = Z/N Z, where N is the period and T {n} = {n + 1}, {n} ∈ Z/N Z. Then with almost every ω ∈ Ω we associate a self-adjoint 2d + 1-diagonal operator J(ω) as follows: (J(ω)x)n =

d

(k)

qn (ω)xn+k ,

k=−d

2 x = {xn }∞ n=−∞ ∈ l (Z),

(k)

where qn (ω) := q (k) (T n ω) and q (−k) (ω) := q (k) (T −k ω). Note that the structure of J(ω) is described by the following identity J(ω)S = SJ(T ω),

(1)

2

where S is the shift operator in l (Z). The last relation indicates strongly that one can associate with the family of matrices {J(ω)}ω∈Ω a natural pair of commuting operators (in the periodic case one just uses the fact that J and S N commute). Namely, let L2dχ(l2 (Z)) be the space of l2 (Z)-valued vector functions, x(ω) ∈ l2 (Z), with the norm ||x||2 = ||x(ω)||2 dχ. Ω

Define

* (Jx)(ω) = J(ω)x(ω),

Then (1) implies

* (Sx)(ω) = Sx(T ω),

x ∈ L2dχ (l2 (Z)).

* * (J*Sx)(ω) = J(ω)Sx(T ω) = SJ(T ω)x(T ω) = (S*Jx)(ω).

Further, S* is a unitary operator and J* is self-adjoint. The space L2dχ(l2 (Z+ )) * It is not invariant with respect to J* but it is invariant is an invariant subspace for S. with respect to J*S*d . Put * 2dχ (l2 (Z+ )), S*+ = S|L

(J*S*d )+ = J*S*d |L2dχ(l2 (Z+ )).

Definition 1.1 (local functional model). We say that a pair of commuting operators A1 : H → H and A2 : H → H has a (local) functional model if there is a unitary embedding i : H → HO in a space HO of functions F (ζ) holomorphic in some domain O with a reproducing kernel (F → F (ζ0 ), ζ0 ∈ O, is a bounded functional

Finite Difference Operators with a Finite-band Spectrum

349

in HO ) such that the operators i⋆ A1 and i⋆ A2 become a pair of operators of multiplication by holomorphic functions, say A1 x → a1 (ζ)F (ζ),

As usual i⋆ A is generated by the diagram H ⏐ ⏐ iF

HO

A2 x → a2 (ζ)F (ζ).

A

−→ i A

H ⏐ ⏐ iF

⋆ −→ HO

Existence of a local functional model implies a number of quite strong consequences. In what follows a(ζ) and b(ζ) denote the functions (symbols) related to the operators (J*S*d )+ and S*+ . Let kζ be the reproducing kernel in HO and let kˆζ be its preimage i−1 kζ in L2dχ(l2 (Z+ )). Then ∗ˆ S*+ kζ , x = kˆζ , S*+ x = kζ , bF .

By the reproducing property Hence,

kζ , bF = b(ζ)F (ζ) = b(ζ)kζ , F .

∗ˆ S*+ kζ , x = b(ζ)kˆζ , x . ∗ ˆ * That is kζ is an eigenvector of S+ with the eigenvalue b(ζ). In the same way, kˆζ is * ∗+ with the eigenvalue a(ζ). an eigenvector of (J*S) Thus, if a functional model exists then the spectral problem / ∗ˆ kζ = b(ζ)kˆζ S*+ , (2) (J*S*d )∗+ kˆζ = a(ζ)kˆζ

has a solution kˆζ antiholomorphic in ζ. Moreover, linear combinations of all kˆζ are dense in L2dχ (l2 (Z+ )). Vice versa, if (2) has a solution of such kind then we define F (ζ) := x, kˆζ ,

||F ||2 := ||x||2 .

This provides a local functional model for the pair S*+ , (J*S*d )+ . The following proposition is evident.

Proposition 1.2. Let U : L2dχ → L2dχ be the unitary operator associated with the ergodic transformation T : (Uc)(ω) = c(T ω), c ∈ L2dχ . We denote by the same letter q both a function q ∈ L∞ dχ and the multiplication operator q (e.g., (qc)(ω) := q(ω)c(ω)). Problem (2) is equivalent to the following spectral problem / d 2 U k q (k) bk (ζ) cζ = z(ζ)cζ , (3) k=−d

where z(ζ) := a(ζ)/bd (ζ) and cζ is an anti-holomorphic L2dχ -valued vector function. Moreover {cζ } is complete in L2dχ if and only if {kˆζ } is complete in L2dχ (l2 (Z+ )).

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We may hope to get a global functional model, see Section 2, on a Riemann surface X0 = {(z, b) : |b| < 1}

with a local coordinates ζ given by (3). We do not claim that the global model always exists (even existence of a local model requires some additional assumptions on the ergodic map and the coefficients functions). But, in particular, in the periodic case, when L2dχ = CN , we have ⎤ ⎡ 0 1 ⎡ ⎤ ⎡ ⎤ c0 c1 ⎥ ⎢ .. . ⎥ ⎢ . .. ⎢ ⎥ ⎢ ⎥ ⎥, U ⎣ ... ⎦ = ⎣ ... ⎦ , U = ⎢ ⎥ ⎢ . . ⎦ ⎣ . 1 cN −1 c0 1 0

where cn := c({n}) and the q (k) ’s became the diagonal matrices. Thus (3) means that the N × N matrix has a nontrivial annihilating vector and we arrive at the curve X0 in the form: 9 d : det U k q (k) bk − zI = 0 k=−d

and the restriction |b| < 1. The surface X0 is in generic case of infinite genus. However we can reduce it because X0 possesses a family of automorphisms. Assume that X0 = D/Γ0 is its uniformization with a Fuchsian group Γ0 . Let e{γ} be an eigenvector of U with an eigenvalue µ ¯ {γ} . The systems of eigenfunctions and eigenvalues form both Abelian groups with respect to multiplication. Using (3) we get immediately that {γ} : (z, b) → (z, µ{γ} b) is an automorphism of X0 . Taking a quotient of X0 with respect to these automorphisms we obtain a much smaller surface X = D/Γ, Γ = {γΓ0 }. In periodic case this means that (1.1) is actually a polynomial expression in z, λ and λ−1 where λ := bN (since it is invariant with respect to the substitutions k b → e2πi N b, 0 < k < N ), see [18], see also Section 6. Note that z is still a function on X but b becomes a character automorphic function. Finally, using z we may glue the boundary of X, remove punched points (where z = ∞) and get in this way a compact Riemann surface Xc , such that X = (Xc \ {P : z(P ) = ∞}) \ E. The simplest assumption is that the boundary E is a finite system of cuts on Xc . In this case we get that the triple {Xc , z, E} characterizes the spectrum of a finite difference operator. We came to this triple basically due to heuristic arguments, but the opposite direction is already a solid mathematical fact: every triple of this kind gives rise to a family of ergodic finite difference operators [18, Sect. 6]. In Sections 2 and 3 we give details on constructions of such operators using the theory of Hardy spaces on Riemann surfaces.


351

Now we would like to note that one can describe all triples of a given type up to a natural equivalence relation. This natural parametrization of the triples is one of the main point of the current paper. It allows us to put consideration from the pure algebraic points of view to a wider setting and to recruit very much analytic tools. 1.2. Parametrization of the spectral curves in terms of branching divisors Probably the best-known result in spectral theory of the nature we want to discuss deals with the description of the spectrum of periodic Jacobi matrices, see, e.g., [17, 27]. Such a set E should have a form of an inverse polynomial image E = T −1 ([−1, 1]). The polynomial T should have all critical points real {ck : T ′ (ck ) = 0} ⊂ R, moreover all critical values tk := T (ck ) should have modulus not less than 1 and their signs should alternate, i.e.: |T (ck )| ≥ 1 and T (ck−1 )T (ck ) < 0

for · · · < ck−1 < ck < · · · .

The claim is that the system of numbers {tk : |tk | ≥ 1, tk−1 tk < 0}, determines a polynomial with the prescribed and ordered critical values tk uniquely, modulo a change of the independent variable z → az + b, a > 0, b ∈ R. The proof uses a special representation of T : T (z) = ± cos φ(z),

where φ is the conformal map of the upper half-plane onto the half-strip with a system of cuts: Π = {w = u + iv : 0 ≤ v ≤ πd} \ ∪k {v = πk, u ≤ hk },

where d = deg T and cosh(hk ) := |tk |, φ(∞) = ∞. We mention a more general theorem of Maclane [15] and Vinberg [28] on the existence and uniqueness of real polynomials (actually, and entire functions) with prescribed (ordered!) sequences of critical values. In the case of polynomials, this theorem says that there is a one to one correspondence between finite “up-down” real sequences · · · ≤ tk−1 ≥ tk ≤ tk+1 ≥ · · · , and real polynomials whose all critical points are real, also, modulo a change of the independent variable z → az + b, a > 0, b ∈ R. The MacLane–Vinberg theorem is based on an explicit description of the Riemann surfaces spread over the plane of the inverse functions T −1 . To be more precise in the general case we start with the following Definition 1.3. We say that two triples (Xc1 , z1 , E1 ) and (Xc2 , z2 , E2 ) are equivalent if there exists a holomorphic homeomorphism h : Xc1 → Xc2 such that z1 = h∗ (z2 ) and E2 = h(E1 ). Note, that for any triple (Xc , z, E) the holomorphic function z : Xc → C, C = C ∪ {∞}, is a ramified covering of C. The fact is that it is possible and very convenient to describe equivalence classes of ramified coverings in terms of dz |P = 0 where ζ is a branching divisor. Namely, a point P ∈ Xc such that dζ

352


local holomorphic coordinate in a neighborhood of P is called ramification point (or, critical point). Its image z(P ) is called a branching point (or, critical value). The set of all branching points of function z form a branching divisor of z. Note that infinity also can be a branching point. Since in our consideration it plays an exclusive role it is convenient in what follows to denote by Z := {zi }N i=1 all branching points from the finite part of the complex plane C. Clearly, branching divisors of equivalent functions are the same. Moreover, the compact holomorphic curve Xc is also uniquely determined by the branching divisor and some additional ramification data (of combinatorial type). Namely, assume that z has degree d. Let w0 ∈ C be a non branching point. Fix a system of non-intersecting paths γ = {γi }i=1,...,N . The ith path γi connects w0 and zi . We want to construct a system of loops li ⊂ C. To construct li we start from w0 and follow first γi almost to zi , then encircle zi counterclockwise along a small circle and finally go back to w0 along −γi . Using li we associate with each branching point an element of the permutation group σi ∈ Σd . The point w0 has exactly d preimages. Let us label them by integers {1,...,d}. Let us follow the loop li and lift this loop to Xc starting from each of the preimages of w0 . The monodromy along path li gives us a permutation σi ∈ Σd of preimages. We add to this system of loops one more l∞ , related to infinity (starting from w0 we go sufficiently far then make a big circle in the clockwise direction and go back to w0 ). This loop gives us one more permutation σ∞ . Note that the product σ1 ·· · ··σN times σ∞ is the identity operator. Therefore, the function z, including its behavior at infinity, determines N branching points and N permutations. These permutations are not uniquely defined, they depend on the labeling of preimages of w0 . Therefore, they are determined, up to a conjugacy by the elements of Σd . Given a set of branching points Z(z) = {zi }i=1,...,N ⊂ C and a system of permutations σ(z) = (σ1 , . . . , σN ) ∈ Σd × · · · × Σd /Σd , where the last quotient is taken with respect to diagonal conjugation, we can restore by Riemann theorem the surface Xc and the function z. Actually there is one more topological condition: the surface should be connected. Throughout the paper we assume that the system of permutations guaranteed this condition to be hold. Hence, the triple (Xc , z, E) is equivalent to the triple (Z, σγ , E). We use this triple as free parameters determining the spectral surface of a 2d + 1-diagonal matrix. Summary. Comparably with the case of Jacobi matrices, where we have only system of cuts (spectral intervals) in the complex plane, in the multidiagonal case we have a new additional system of parameters. We have to fix in C a system of critical points Z, and associate to them a system of permutation σ, which actually depends on the base point w0 ∈ C \ Z and the system of paths γ. They define a Riemann surfaces Xc and a covering z. Then on the set z −1 (R) ⊂ Xc we chose a system of cuts E, and thus (Xc , z, E) is restored up to the equivalence 1.3.


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Remark 1.4. As it was mentioned by the branching divisors language one can extend consideration from the pure algebraic level. In particular, we are very interested in infinite-dimensional generalizations: the point is that starting from the 5-diagonal case we have the branching divisor Z as a completely new system of parameters characterizing the spectral surface. What is its influence on the properties of the corresponding surface? Consider, indeed, the simplest case of 5-diagonal matrices. Then we have

just to specify the point set Z (all permutations are of 1 2 ). What is the speed of accumulation of an infinite system the form σj = 2 1 of points Z to, say, a finite system of cuts E so that Xc \ E is of Widom type, or of Widom type with the Direct Cauchy Theorem (for the definition of these type of surfaces see [12], see also [26] for their role in the spectral theory)? We provide here the following example. Let W (z) be the infinite Blaschke product in the upper half-plane with zeros at {zk }k≥0 . Define C

R = {P = (z, w) : w2 = W,

ℑz ≥ 0}.

z−z0 z−¯ z0

is the Green function in R. Note that Ck = (zk , 0), k ≥ 1 are its Then critical points, and therefore the Widom function is 8 z − z¯0 ∆= W . z − z0

Note that the Carleson condition for this function on R is the standard Carleson condition for {zk }k≥1 in the upper half-plane. Thus the Blaschke and Carleson conditions on the zero set in the half-plane guarantee that R is of Widom type and of Widom type with the Direct Cauchy Theorem respectively. 1.3. Structure of the paper and main results In Section 2, having the spectral surface fixed, we define a system of Hardy spaces on it (natural counterparts of the Hardy space in the unit disk). In that section we assume that there is only one “infinity” P0 ∈ Xc \ E, z(P0 ) = ∞, that is, the product of permutations σ1 · · · · · σN is a cycle. There is an intrinsic basis in the Hardy space: each next basis elements has at P0 a zero of bigger and bigger multiplicity. Of course, this is a counterpart of the standard basis system {ζ n }n≥0 in the standard H 2 . Extending this basis to the negative integers (the system extends in the direction of functions having a pole at P0 with growing multiplicity) one gets a basis in the whole L2 . Finally the multiplication operator, by the covering map with respect to this basis, is the 2d + 1-diagonal matrix (to this end it is important to note that z has a pole of multiplicity d at P0 ). The complex Green function with zero at P0 is playing the role of the symbol of the shift operator. Then we study the question of uniqueness of such a model for an ergodic operator (see Theorem 2.3 and the example right after it). The general case (several “infinities”) is considered in Section 3. Now the d-root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface becomes the symbol of the shift. We

354


have the ordering of the infinities as one more parameter, defining the basis system and the corresponding multidiagonal matrix. In Section 4 we demonstrate, how an application of our general construction to a particular covering, generate the well-known and now widely discussed almost periodic CMV matrices. Starting from Section 5 we discuss an important theme: covering of one spectral surface by another one and the related transformations on the set of multidiagonal operators (so-called Renormalization Equations). Let π : Yc → Xc be a d-sheeted covering. For a system of cuts E define F = π −1 (E). Then we have π : Yc \ F → Xc \ E. The study is based on the relation between the Hardy spaces on these Riemann surfaces. The Renormalization Equations generated by polynomial coverings have played an important role in studying of almost periodic Jacobi matrices with a singular continuous spectrum, [5], see also [21], [22]. They act in the most natural way on periodic Jacobi matrices, see Section 6. In Section 7 we proof several new results dealing with Renormalization Equations for periodic Jacobi matrices: describe the complete set of their solutions; show their relation with the Ruelle operators. Finally, we give a possible generalization of the constructions from Section 6 for a wider class of almost periodic Jacobi matrices with a singular continuous spectrum. In particular, we prove the Lipschitz property of the Darboux transform. Having in mind importance of the paper [3], where the Renormalization Equation generated by “just” quadratic polynomial was used, we investigate in Section 8 the case of rational double covering π(v) = τ v − τ −1 v , τ > 1. As usual, the renormalization procedure is simpler to formulate for operators acting on the (integer) half-axis Z+ . 2 = l2 (Z+ ) with a Definition 1.5. Let A be a self-adjoint operator acting in l+ cyclic vector |0 and the spectrum on [−1, 1]. We define its transform π ∗ (A) in the following steps. First we define the upper triangular matrix Φ (with positive diagonal entries) by the condition

A2 + 4τ (τ − 1) = Φ∗ Φ. Then we introduce A∗ := ΦAΦ−1 and define the operator # $ A Φ∗ , Φ A∗ 2 2 2 2 2 acting in l+ ⊕ l+ . Finally, using the unitary operator U : l+ → l+ ⊕ l+ , such that

U |2k = |k ⊕ 0, we construct π ∗ (A) :=

1 ∗ U 2τ

#

U |2k + 1 = 0 ⊕ |k A Φ∗ Φ A∗

$

2 2 U : l+ → l+ .


355

We give a theorem (see Theorem 8.8) on the weak convergence of the iterative procedure An+1 = π ∗ (An ) to an operator with a simple singular continuous spectrum supported on the Julia set of the given expanding mapping and pose here a question on the contractivity of the renormalization operator, at least for large values of τ . The main general conjecture deals with contractivity of all renormalizations, generated by a covering with sufficiently large critical values.

2. The Global Functional Model (single infinity case). Uniqueness Theorem 2.1. Hardy spaces and bases There are different ways to define Hardy spaces on Riemann surfaces, – the spaces of vector bundles, multivalued functions or forms. These definitions are equivalent. We start from 1-forms, the most natural object with this respect from our point of view. Let π(ζ) : D → X be a uniformization of the surface X = Xc \ E. Thus there exists a discrete subgroup Γ of the group SU (1, 1) consisting of elements of the form $ # γ12 γ , γ11 = γ22 , γ12 = γ21 , det γ = 1, γ = 11 γ21 γ22 such that π(ζ) is automorphic with respect to Γ, i.e., π(γ(ζ)) = π(ζ), ∀γ ∈ Γ, and any two preimages of P ∈ X are Γ-equivalent. We normalize Z(ζ) := (z ◦ π)(ζ)

by the conditions Z(0) = ∞, (ζ d Z)(0) > 0. Note that Γ acts dissipatively on T with respect to the Lebesgue measure dm, that is there exists a measurable (fundamental) set E, which does not contain any two Γ-equivalent points, and the union ∪γ∈Γ γ(E) is a set of full measure. In fact E can be chosen as a finite union of intervals, – the T-part of the boundary of the fundamental domain. For the space of square summable functions on E (with respect to dm), we use the notation L2dm|E. A character of Γ is a complex-valued function α : Γ → T, satisfying α(γ1 γ2 ) = α(γ1 )α(γ2 ),

γ1 , γ2 ∈ Γ.

The characters form an Abelian compact group denoted by Γ∗ . The further Hardy spaces on X are marked by characters of Γ. Let f be an analytic function in D, γ ∈ Γ. Then we put f |[γ]k =

f (γ(ζ)) (γ21 ζ + γ22 )k

k = 1, 2.

Notice that f |[γ]2 = f for all γ ∈ Γ, means that the form f (ζ)dζ is invariant with respect to the substitutions ζ → γ(ζ) (f (ζ)dζ is an Abelian integral on D/Γ). Analogously, f |[γ] = α(γ)f for all γ ∈ Γ, α ∈ Γ∗ , means that the form |f (ζ)|2 |dζ| is invariant with respect to these substitutions.

356


We recall, that a function f (ζ) is of Smirnov class, if it can be represented as a ratio of two functions from H ∞ with an outer denominator. The following spaces related to the Riemann surface D/Γ are counterparts of the standard Hardy spaces H 2 (H 1 ) on the unit disk. Definition 2.1. The space A21 (Γ, α) (A12 (Γ, α)) is formed by functions f , which are analytic on D and satisfy the following three conditions 1) f is of Smirnov class 2) f |[γ] = α(γ)f (f |[γ]2 = α(γ)f ) ∀γ ∈ Γ

|f | dm < ∞ . |f |2 dm < ∞ 3) E

E

A21 (Γ, α) is a Hilbert space with the reproducing kernel k α (ζ, ζ0 ), moreover 0 < inf∗ k α (ζ0 , ζ0 ) ≤ sup k α (ζ0 , ζ0 ) < ∞. α∈Γ

Put

α∈Γ∗

k α (ζ) . k α (ζ) = k α (ζ, 0) and K α (ζ) = Kζα (0) = k α (0) We need one more special function. The Blaschke product γ(0) − ζ |γ(0)| b(ζ) = ζ 1 − γ(0)ζ γ(0) γ∈Γ,γ =12

is called the Green’s function of Γ with respect to the origin. It is a characterautomorphic function, i.e., there exists µ ∈ Γ∗ such that b(γ(ζ)) = µ(γ)b(ζ).

(4)

Note, if G(P ) = G(P, P0 ) denotes the Green’s function of the surface X, then G(π(ζ)) = − log |b(ζ)|.

We are ready to construct the basis in A21 (Γ, α). Consider the following subspace of this space {f ∈ A21 (Γ, α) : f (0) = 0}.

The following two facts are evident

1) {f ∈ A21 (Γ, α) : f (0) = 0} = {bf˜ : f˜ ∈ A21 (Γ, µ−1 α)} = bA21 (Γ, µ−1 α), 2) A21 (Γ, α) = {K α } ⊕ {f ∈ A21 (Γ, α) : f (0) = 0}. Thus A21 (Γ, α) ={K α } ⊕ bA21 (Γ, µ−1 α) −1

={K α } ⊕ {bK αμ } ⊕ b2 A21 (Γ, µ−2 α),

and so on. Basically we proved the following theorem, note, however, that the second statement is not a direct consequence of the first one.


357

−n

Theorem 2.2. Given α ∈ Γ∗ , the system of functions {bn K αμ }n≥0 forms an −n orthonormal basis in A21 (Γ, α); the system {bn K αμ }n∈Z is an orthonormal basis 2 in Ldm|E. 2.2. The Global Functional Model Of course the constructions in this section and our speculations in Section 1 are closely related and of mutual influence. In this subsection we close the construction from Section 2.1 by proving the Global Functional Model Theorem. Let Γ0 := ker µ, µ given by (4), that is Γ0 = {γ ∈ Γ : µ(γ) = 1}. Evidently, b(ζ) and (zbd )(ζ) are holomorphic functions on the surface X0 = D/Γ0 . Assume that α0 ∈ Γ0 can be extended to a character on Γ, i.e., Ωα0 = {α ∈ Γ∗ : α|Γ0 = α0 } = ∅. Note that the set of characters Ωι = {α ∈ Γ∗ : α|Γ0 = ι} where ι(γ) = 1 for all γ ∈ Γ0 is isomorphic to the set (Γ/Γ0 )∗ . Let us fix an element α ˆ 0 ∈ Ωα0 . Since {α ∈ Γ∗ : α|Γ0 = α0 } = {α ˆ 0 β : β ∈ Γ∗ such that β|Γ0 = ι} we can define a measure dχα0 (α) on Ωα0 by the relation dχα0 (α) = dχα0 (ˆ α0 β) = dχι (β), where dχι (β) is the Haar measure on (Γ/Γ0 )∗ (the measure dχα0 (α) does not depend on the choice of the element α ˆ0 ). Obviously, T α := µ−1 α is an invertible ergodic measure-preserving transformation on Ω = Ωα0 with respect to the measure dχ = dχα0 . The following theorem is a slightly modified version of Theorem 2.2 given in [29]. Theorem 2.3. With respect to the basis from Theorem 2.2, the multiplication operator by z is a 2d + 1-diagonal ergodic finite difference operator with Ω = Ωα0 , dχ = dχα0 , T α := µ−1 α and α0 = α|Γ0 . Moreover, the operators S*+ and (J*S*d )+ are unitary equivalent to multiplication by b and (bd Z) in A21 (Γ0 , α0 ) respectively. This unitary map is given by the formula −n xn (α)bn K αμ , f ∈ A21 (Γ0 , α0 ), f |[γ]α−1 (γ) = {γ}∈Γ/Γ0

n∈Z+

where the vector function x(α) := {xn (α)} belongs to L2dχ(l2 (Z+ )). 2.3. Uniqueness Theorem The natural question arises: up to which extend is our functional realization unique?

358


Theorem 2.4. Assume that a finite difference ergodic operator has a finite band functional model that is there exist a triple {Xc , z˜, E}, a character α0 ∈ Γ∗ and a ˜ ˜ := Ωα0 such that F T ω = µ−1 F ω, χ(F −1 (A)) = χ(A), ˜ A ⊂ Ω, map F from Ω to Ω with dχ ˜ := dχα0 , here µ is the character of the Green’s function ˜b on Xc \ E. Moreover q (k) (ω) = q˜(k) (F ω), where the coefficients q˜(k) (α) are generated by the −n multiplication operator z˜ with respect to the orthonormal basis {˜bn K αμ }n∈Z . ˜ If the functions z˜ and {d log b/d˜ z } separate points on Xc \ E then any local functional model is generated by one of the branches of the function ˜b. For the proof see [23]. The following example shows that in the case when these two functions z˜ and {d log ˜b/d˜ z } do not separate points on Xc \ E one can give different global functional realizations for the same ergodic operator. Example [23]. Let J = S d + S −d . There exist a “trivial” functional model with Xc \ E ∼ D. In this case J is the multiplication operator by z = ζ d + ζ −d with respect to the standard basis {ζ l } in L2T . Note that b = ζ, thus w :=

d log b 1 1 = d , dz ζ − ζ −d d

that is z 2 + (wd)−2 = 4, |(wd)−1 + z| < 2. On the other hand let us fix any polynomial T (u), deg T = d, with real critical values on R \ [−2, 2] and define Xc \ E = T −1 (C \ [−2, 2]) ∼ C \ T −1 [−2, 2]. As we discussed the last set is the resolvent set for a d-periodic Jacobi matrix, say J0 . Moreover T (J0 ) = J, and − log |b| is just the Green’s function of this domain in the complex plane. So, using the standard functional model for J0 (see Section 7 for details) with the symbols u and b we get a functional model for J with z = T (u) and the same b. Note that as before z 2 + (wd)−2 = 4, |(wd)−1 + z| < 2 b with w := d log dz . Remark 2.5. In [8] the identity T (J0 ) = S d +S −d which holds for a periodic Jacobi matrix J0 with spectrum T −1 [−2, 2] is called the Magic Formula. There it plays an important role in proving counterparts of Denisov–Rakhmanov and Killip–Simon Theorems for perturbations of periodic Jacbi matrices.

3. Several infinities case. Existence Theorem Now we examine the situation in which the reduced surface Xc has several “infinities”, that is, the covering function z (the symbol of an almost periodic operator) equals infinity at several (distinct) points on Xc . We start with a simple example.


359

3.1. A five diagonal matrix of period two Assume that z is a two sheeted covering with only two branching points, say, 1 2 z1 = −2, z2 = 2. Thus the corresponding substitutions are σ1 = σ2 = . In 2 1 this case Xc is equivalent to the complex plane C, moreover we can put 1 (5) z = v + , v ∈ C. v Thus, on this surface we have two “infinities” v = ∞ and v = 0. Note that z −1 (R) = T ∪ R. We cut C over the interval [a, b], where 0 < a < b < 1, that is we consider Xc \ E of the form C \ [a, b]. We uniformize Xc \ E by a + b b − a 1/ζ + ζ + , ζ ∈ D. 2 2 2 For v = ∞ we have ζ0 = 0. Solving b+a ζ +1=0 ζ2 + 2 b−a v=

√

√

a √ . As a result we get a we get the image of the second infinity in D, ζ1 = − √b− b+ a symbol function z : D → C (see (5), (3.1)) for the forthcoming operator with two infinities ζ0 , ζ1 . Next we derive the symbol b for the shift operator. Recall that the Green function in D is the Blaschke factor ζ − ζ1 bζ0 = ζ, bζ1 = . 1 − ζζ1 The product bζ0 bζ1 is the smallest unimodular multiplier that cancels poles of z. Since b2 is of the same nature, b2 z is holomorphic with a unimodular function b on T, we have b2 = bζ0 bζ1 . Finally, we need a certain functional space and an intrinsic basis in it that generalize the construction in Theorem 2.2. Recall b should be related to the shift S, and we are going to define the periodic operator J as the multiplication operator with respect to this basis. To this end we define the following functional spaces. Given αk ∈ T, k = 0, 1, we associate the space H 2 (α0 , α1 ) of analytic multivalued functions f (ζ), ζ ∈ D \ {ζ0 , ζ1 }, such that |f (ζ)|2 has a harmonic majorant and

f ◦ γi = αi f,

where γi is a small circle around ζi . Such a space can be reduced to the standard Hardy space H 2 , moreover H 2 (α0 , α1 ) = bτζ00 bτζ11 H 2 ,

αk = e2πiτk , 0 ≤ τk < 1.

Lemma 3.1. The space bH 2 (−1, 1) is a subspace of H 2 (1, −1) having a one-dimensional orthogonal complement, moreover (6) H 2 (1, −1) = { bζ1 kζ0 } ⊕ bH 2 (−1, 1), where kζi is the reproducing kernel of the standard H 2 with respect to ζi .

360


This lemma allows us to repeat the construction of Subsection 2.1. Iterating, now, the decomposition (6) H 2 (1, −1) ={ bζ1 kζ0 } ⊕ bH 2 (−1, 1) ={ bζ1 kζ0 } ⊕ b{ bζ0 kζ1 } ⊕ b2 H 2 (1, −1) = · · · ,

one gets an orthogonal basis in H 2 (1, −1) consisting of vectors of two sorts b2m { bζ1 kζ0 } and b2m+1 { bζ0 kζ1 }.

Note that this orthogonal system can be extended to negative integers m so that we obtain a basis in L2 (1, −1).

Theorem 3.2. With respect to the orthonormal basis / 2m k bζ1 ||kζζ0 || , n = 2m b 0 en = k 2m+1 b bζ0 ||kζζ1 || , n = 2m + 1 1

the multiplication operator by z is a 5-diagonal matrix of period 2. 3.2. General case Let z : Xc → C be d-sheeted covering with d (distinct) infinities. Further, let D/Γ be a uniformization of Xc \ E. For a given character α ∈ Γ∗ and a fixed ordering of infinities P1 , P2 , . . . , Pd we denote by • bj = bPj the Green function with respect to Pj , bj ◦ γ = µj (γ)bj , µj ∈ Γ∗ ; • kjα = kPαj the reproducing kernel of A21 (Γ, α) with respect to Pj , Kjα :=

kjα . ||kjα ||

Similar to the previous subsection we get 1

Theorem 3.3. Let b = (b1 . . . bd ) d . With respect to the orthonormal basis ⎧ d−1 1 α(μ ...μ )−m ⎪ d ⎪bdm b1d . . . bd−1 Kd 1 d , n = dm ⎪ ⎪ d−1 1 ⎪ αμ−1 (μ ...μ )−m ⎨ dm+1 d b b 2 . . . bd d K 1 1 1 d , n = dm + 1 en = ⎪ ⎪ ⎪. . . ⎪ −1 −m ⎪ αμ−1 1 ...μd−1 (μ1 ...μd ) ⎩bdm+d−1 b d1 . . . b d−1 d , n = dm + d − 1 d d−2 Kd−1

(7)

the multiplication operator by z is a (2d + 1)-diagonal almost periodic matrix.

4. Five-diagonal almost periodic self-adjoint matrices and OPUC We start again with the two-sheeted covering (5). We have Xc ≃ C and z −1 (R) = R ∪ T, but let us, in this case, cut C on an arbitrary finite union of (necessary non-degenerate) arcs on the unit circle, E ≃ {T \ ∪lj=0 (aj , bj )}.


361

The domain Xc \ E is conformally equivalent to the quotient of the unit disk by the action of a discrete group Γ = Γ(E). Let v : D → {C \ T} ∪ {∪lj=0 (aj , bj )}

(8)

be a covering map, v ◦ γ = v, ∀γ ∈ Γ. In what follows we assume the following normalization to be hold v : (−1, 1) → (a0 , b0 ), so that one can choose a fundamental domain F and a system of generators {γj }lj=1 of Γ such that they are symmetric with respect to the complex conjugation: F = F,

γj = γj−1 .

Denote by ζ0 ∈ F the preimage of the origin, v(ζ0 ) = 0, then v(ζ0 ) = ∞. Thus the infinities of z defined by (5) and (8) on D/Γ are trajectories of ζ0 and ζ0 under the action of the group Γ, P0 = {γ(ζ0 )}γ∈Γ and P1 = {γ(ζ0 )}γ∈Γ . There are two infinities that we have in the case under consideration. Thus, to define the function b (the symbol of the shift operator) we have to introduce the Green functions B(ζ, ζ0 ) and B(ζ, ζ0 ). It is convenient to normalize them by B(ζ0 , ζ0 ) > 0 and B(ζ0 , ζ0 ) > 0. Then v(ζ) = eic

B(ζ, ζ0 ) . B(ζ, ζ0 )

(9)

Also, we can rotate (if necessary) the v-plane so that c = 0. Note that B(ζ, ζ0 ) is a character-automorphic function B(γ(ζ), ζ0 ) = µ(γ)B(ζ, ζ0 ), ∗

γ ∈ Γ,

with a certain µ ∈ Γ . By (9), B(ζ, ζ0 ) has the same factor of automorphy, B(γ(ζ), ζ0 ) = µ(γ)B(ζ, ζ0 ), 2

γ ∈ Γ.

By the definition b = B(ζ, ζ0 )B(ζ, ζ0 ) we get a multivalued analytic function b on the punched surface {D/Γ} \ {P0 , P1 }. In this case we have only two possibilities for ordering of the infinities: {P0 , P1 } and {P1 , P0 }. According to Theorem 3.3, to any of them, say the first one, and to an arbitrary α ∈ Γ∗ we can associate the operator J = J(−1, 1; α) by 2 fixing the space H 2 (−1, 1; α) = B(ζ, ζ0 )A1 (α)2 and a natural basis in it. Up to a common multiplier B(ζ, ζ0 ) it is a basis in A1 (α) of the form −1

−2

K α (ζ, ζ0 ), B(ζ, ζ0 )K αμ (ζ, ζ0 ), B(ζ, ζ0 )B(ζ, ζ0 )K αμ (ζ, ζ0 ), . . . .

Thus we get the same system of functions that we used describing almost periodic Verblunsky coefficients [20]. The last one we can define by a(α) = a =

K α (ζ0 , ζ0 ) . K α (ζ0 , ζ0 )

In [20] they appear in the following recursion −1

K α (ζ, ζ0 ) = a(α)K α (ζ, ζ0 ) + ρ(α)B(ζ, ζ0 )K αμ (ζ, ζ0 ), −1 K α (ζ, ζ0 ) = a(α)K α (ζ, ζ0 ) + ρ(α)B(ζ, ζ0 )K αμ (ζ, ζ0 ),

(10)

362


where

−1

K αμ (ζ0 , ζ0 ) . K α (ζ0 , ζ0 ) The goal of this section is to represent J in terms of Verblunsky coefficients. ρ(α) = ρ =

1 − |a|2 = B(ζ0 , ζ0 )

Lemma 4.1. With respect to the basis −1

. . . , K αμ (ζ, ζ0 )/B(ζ, ζ0 ), K α (ζ, ζ0 ), B(ζ, ζ0 )K αμ (ζ, ζ0 ), . . .

(11)

the multiplication operator by v is a matrix having at most two non-vanishing diagonals over the main diagonal. Moreover, ⎤ ⎡ ρ(αµ)ρ(αµ2 ) 0 ⎥ ⎢. . ⎥ ⎢ . −ρ(αµ)a(αµ2 ) 0 ⎥ ⎢ −1 ⎥. ⎢ ρ(α)a(αµ ) −a(αµ)a(α) (12) v∼⎢ ⎥ ⎥ ⎢ −1 ) ∗ −a(α)a(αµ ⎦ ⎣ .. . Similarly, the multiplication operator by 1/v is of the form ⎡ ⎤ 0 0 ⎢. . ⎥ ⎢ . ρ(αµ)a(α) ⎥ ρ(α)ρ(αµ) ⎢ ⎥ ⎢ ⎥. −a(αµ)a(α) −ρ(α)a(αµ) 1/v ∼ ⎢ ⎥ −1 ⎢ ⎥ )a(α) ∗ −a(αµ ⎣ ⎦ .. .

(13)

Proof. We give a proof, say, for (12). Recall (9), from which we can see that the decomposition of the vector v(ζ)K α (ζ, ζ0 ) begins with 2

v(ζ)K α (ζ, ζ0 ) = c0

K αμ (ζ, ζ0 ) K αμ (ζ, ζ0 ) + c2 K α (ζ, ζ0 ) + · · · . + c1 B(ζ, ζ0 ) B(ζ, ζ0 )B(ζ, ζ0 )

Multiplying by the denominator B(ζ, ζ0 )B(ζ, ζ0 ) we get 2

B 2 (ζ, ζ0 )K α (ζ, ζ0 ) =c0 K αμ (ζ, ζ0 ) + c1 K αμ (ζ, ζ0 )B(ζ, ζ0 ) +c2 K α (ζ, ζ0 )B(ζ, ζ0 )B(ζ, ζ0 ) + · · · . First we put ζ = ζ0 . By the definition of ρ(α) we have c0 = B 2 (ζ0 , ζ0 )

K α (ζ0 , ζ0 ) = ρ(αµ)ρ(αµ2 ). K αμ2 (ζ0 , ζ0 )

Putting ζ = ζ0 in (14) and using the definition of a(α), we have c1 = −c0

2 K αμ (ζ0 , ζ0 ) αμ K (ζ0 , ζ0 )B(ζ0 , ζ0 )

= −ρ(αµ)ρ(αµ2 )

a(αµ2 ) = −ρ(αµ)a(αµ2 ). ρ(αµ2 )

(14)


363

Doing in the same way we can find a representation for c2 that would involve derivatives of the reproducing kernels. However, we can find c2 in terms of a and ρ calculating the scalar product c2 = B 2 (ζ, ζ0 )K α (ζ, ζ0 ), B(ζ, ζ0 )B(ζ, ζ0 )K α (ζ, ζ0 ) . Since B(ζ, ζ0 ) is unimodular, using (10), we get c2 =

K αμ (ζ, ζ0 ) − a(αµ)K αμ (ζ, ζ0 ) , B(ζ, ζ0 )K α (ζ, ζ0 ) . ρ(αµ)

Recall that k α (ζ, ζ0 ) = K α (ζ, ζ0 )K α (ζ0 , ζ0 ) is the reproducing kernel. Thus c2 = −

a(αµ) a(αµ) B(ζ0 , ζ0 )K α (ζ0 , ζ0 ) =− ρ(αµ)a(α) = −a(αµ)a(α). ρ(αµ) K αμ (ζ0 , ζ0 ) ρ(αµ) −1

To find the decomposition of the vector v(ζ)B(ζ, ζ0 )K αμ (ζ, ζ0 ) is even simpler since only two leading terms are involved. Note that all other columns of the matrix in (12), starting from these two, can be obtain by the character’s shift by µ−2 along diagonals. Now let us remind the CMV representation for operators related to OPUC [24]. For a given sequence of numbers from D . . . , a−1 , a0 , a1 , a2 , . . .

(15)

define unitary matrices Ak =

#

ak ρk

$ ρk , −ak

ρk =

1 − |ak |2 ,

and unitary operators in l2 (Z) given by block-diagonal matrices ⎤ ⎤ ⎡ ⎡ .. .. . . ⎥ ⎥ ⎢ ⎢ ⎥ −1 ⎥ ⎢ ⎢ A−1 A−2 ⎥S . ⎥ , A1 = S ⎢ A0 = ⎢ ⎥ ⎥ ⎢ ⎢ A A 1 0 ⎦ ⎦ ⎣ ⎣ .. .. . .

The CMV operator A, related to the sequence (15), is the product A = A({ak }) := A0 A1 .

(16)

Theorem 4.2. Define the sequence ak = a(αµ−k ). Then v ∼ A({ak }), see (16). Proof. Note that the operators in (12) and (13) are mutually conjugated, therefore the under-diagonal entries of both operators are also known. The rest is an easy direct computation.

364


Theorem 4.3. With respect to the basis (11) the multiplication operator by z, ⎤ ⎡ q (−2) (α) 0 ⎥ ⎢. . ⎥ ⎢ . q (−1) (α) q (−2) (αµ−1 ) ⎥ ⎢ (0) ⎥, (−1) (αµ−1 )} z∼⎢ q (α) {q ⎥ ⎢ ⎥ ⎢ ∗ q (0) (αµ−1 ) ⎦ ⎣ .. .

is defined by the functions

q (−2) (α) = ρ(αµ)ρ(αµ2 ), and

q (−1) (α) = ρ(αµ){a(α) − a(αµ2 )},

q (0) (α) = −2ℜ{a(α)a(αµ)}.

It is worth to mention that the second column is not only the shift by µ−1 , there is also conjugation. Proof. Recall that z = v + 1/v and use the previous lemma.

Using general constructions from [18] one can define and integrate the flows hierarchy given by ˙ = [(An + A−n )+ , A]. (17) A Note that for n = 1, (17) gives the Schur flow [11].

5. Coverings Discussing this subject we prefer to use a functional version of presentation of Hardy spaces. First we introduce these spaces and then we will remark how they are related to the spaces of forms. Let ωP0 denote the harmonic measure on an open surface Xc \ E with respect 1 d log b(P, P0 ) to P0 ∈ Xc \ E. Note that ωP0 is the restriction of the differential 2πi on E. By H 2 (α, ωP0 ) we denote the closure of H ∞ (α) in L2 with respect to the measure ωP0 . The natural question is how this space is related to the space with another point fixed, say P1 ∈ Xc \ E, or, more generally, with H 2 (α, ω), which denotes the closure of H ∞ (α) with respect to an equivalent norm given by a measure of the form ω = ρωP0 , where 0 < C1 ≤ ρ ≤ C2 < ∞. (By the Harnack Theorem ωP1 and ωP0 satisfy this property.) To answer it, let us define an outer function φ, such that ρ = |φ|2 . This function belongs to H ∞ (β) with a certain β ∈ Γ∗ . In this case f → φf

is a unitary map from H 2 (α, ω) to H 2 (αβ, ωP0 ). Then the equality α α (P ; ω)φ(P )φ(Q) H 2 (αβ,ωP0 ) =f (P ), kQ (P ; ω)φ(Q) H 2 (α,ω) (φf )(P ), kQ

=f (Q)φ(Q)


365

shows that the reproducing kernels of H 2 (αβ, ωP0 ) and H 2 (α, ω) are related by αβ α (P ; ωP0 ) = kQ (P ; ω)φ(P )φ(Q). kQ

For the normalized kernels we have αβ α (P ; ωP0 ) = KQ (P ; ω) KQ

φ(Q) φ(P ). |φ(Q)|

Therefore the matrix of a multiplication operator in H 2 (α, ω), with respect to the reproducing kernels based basis, actually can be obtained by a character’s shift for the matrix with the same symbol related to the chosen space H 2 (α, ωP0 ). (Let us mention in brackets a specific normalization of a basis vector given by the φ(Q) unimodular factor |φ(Q)| .) The relations between A21 (α) and H 2 (α) are of the same nature. Indeed, let ρ : D/Γ → Xc \ E, ρ(0) = P0 , be the uniformization of the given surface. Then H 2 (α) is a subspace of the standard H 2 in D: f (ρ(ζ)) ∈ H 2 for f ∈ H 2 (α), moreover f 2 = |f (ρ(t))|2 dm(t), T

where dm is the Lebesgue measure. Fix a fundamental set E for the action of Γ on T, T = ∪γ∈Γ γ(E). Then |f (ρ(t))|2 |ψ(t)|2 dm(t), f 2 = E

where

|ψ(t)|2 :=

γ∈Γ

|γ ′ (t)|.

Again, we can consider ψ as an outer function and then 2

f → (f ◦ ρ)ψ

is the unitary map from H (α) to A21 (αβ), where the character β is generated by the 1-form ψ. The β here is a particular character, so when α runs on the whole group Γ∗ αβ covers also all characters, and we have one to one correspondence between two ways of writing Hardy spaces. But, as we noted above, working with coverings, it will be convenient to use character automorphic H 2 -spaces with respect to the following specific measure l

ω=

1 ωP , l i=1 i

(18)

associated with a system of points {Pi } on Xc \ E. Naturally in what follows Pi ’s are infinities on Xc \ E. Now we can go back to the coverings. Let ΓX (respectively ΓY ) be the fundamental group on Xc \ E (respectively Yc \ E). We have π∗ : ΓY → ΓX (π∗ (γ) is the image of a contour γ ∈ ΓY ) and π ∗ : ΓX → ΓY (π ∗ (γ) is the full preimage

366


of a contour γ ∈ ΓX ). Note that π ∗ π∗ = d Id. The maps π∗ : Γ∗Y → Γ∗X and π ∗ : Γ∗X → Γ∗Y are defined by duality. For a system of points {Pi }li=1 , Pi ∈ Xc \ E, define the measure ω∗ on E by (i) (18). Let {Qk }dk=1 = π −1 (Pi ). Define ω∗ = In this case

F

1 ωQ(i) . k ld

(f ◦ π)ω ∗ =

E

f ω∗ .

(19)

Moreover,

f ω∗ = F

E

(Lf )ω∗ ,

where (Lf )(P ) =

1 d

f (Q).

(20)

π(Q)=P

As a direct consequence of (19), (20) we get Lemma 5.1. The map V : H 2 (α, ω∗ ) → H 2 (π ∗ α, ω ∗ ), defined by V f = f ◦ π,

f ∈ H 2 (α, ω∗ ),

(21)

is an isometry with (V ∗ f )(P ) = (Lf )(P ),

f ∈ H 2 (π ∗ α, ω ∗ ).

(22)

Also, V kPα0 =

1 d

(0)

∗

π α kQ (0) . j

π(Qj )=P0

for the reproducing kernel kPα0 ∈ H 2 (α, ω∗ ). Theorem 5.2. Let z∗ : Xc \ E → C. Using the notations introduced above, assume that z∗ : E → R,

z∗−1 (∞) ⊂ {Pi }li=1

Let z∗ (α) be the multiplication operator by z∗ with respect to the basis (7). For (i) an arbitrary ordering of {Qk }1≤k≤d,1≤i≤l subordinated to the ordering of {Pi } ∗ ∗ consider the operator z (π α) with the symbol z ∗ := z∗ ◦ π and the, by (21), related isometry V. Then, the following relations hold V∗ z∗ (π ∗ α) = z∗ (α)V∗ ,

V∗ S d = SV∗ .

(23)

Equations (23) are very close to the so-called Renormalization Equations that we start to discuss now.


367

6. The renormalization of periodic matrices We recall some basic facts from the spectral theory of periodic Jacobi matrices. The spectrum E of any periodic matrix J is an inverse polynomial image E = U −1 [−1, 1]

(24)

the polynomial U of degree g + 1 should have all critical points {cU } real and for all critical values |U (cU )| ≥ 1. For simplicity we assume that |U (cU )| > 1. Then the spectrum of J consists of g intervals E = [b0 , a0 ] \ (∪gj=1 (aj , bj )). Also it would be convenient for us to normalize U by a linear change of variable such that b0 = −1 and a0 = 1. In this case U is a so-called expanding polynomial. Having the set E of the above form fixed, let us describe the whole set of periodic Jacobi matrices J(E) with the given spectrum. To this end we associate with U the hyper-elliptic Riemann surface (the surface is given by (1.1) with λ = bN , N = g + 1) X = {Z = (z, λ) : λ − 2U (z) + λ−1 = 0}. The involution on it is denoted by τ , τ Z :=

1 ∈ X. z, λ

(25)

The set X+ = {Z ∈ X : |λ(Z)| < 1} ¯ \ E, in fact, z(Z) ∈ C ¯ \ E if Z ∈ X+ . we call the upper sheet of X. Note X+ ≃ C The following well-known theorem describes J(E) in terms of real divisors on X. The Jacobian variety of X, Jac(X), is a g-dimensional complex torus, Jac(X) ≃ Cg /L(X), where L is a lattice (that can be chosen in the form L = Zg + ΩZg with ℑΩ > 0). Consider the g-dimensional real subtorus consisting of divisors of the form D(E) = {D = D+ − DC , D+ :=

g i=1

Zi : Zi ∈ X, z(Zi ) ∈ [ai , bi ]},

here DC is a point of normalization that we choose of the form DC :=

g i=1

Ci : Ci ∈ X, z(Ci ) = (cU )i , |λ(Ci )| > 1,

– the collections of the points on the lower sheet with the z-coordinates at the critical points. (At least topologically, it is evident D(E) ≃ Rg /Zg .) Theorem 6.1. For given E of the form (24) there exists a one-to-one correspondence between J(E) and D(E).

368


˜ and T be polynomials of the form described above, and let us Let now U ˜ ◦ T . Then we have a covering π of the Riemann surface X ˜ associated define U = U ˜ by the surface X associated to U : to U π(z, λ) = (T (z), λ),

(26)

˜+. note π : X+ → X According to the general theory, this covering generates different natural mappings [16], in particular, ˜ π∗ : Jac(X) → Jac(X), and ˜ → Jac(X). π ∗ : Jac(X)

Thus, in combination with Theorem 6.1, we get the map π∗

T ˜ D( ⏐ ⏐E) −→ D(E) ⏐ ⏐ F F

˜ J(E)

J

T −→ J(E)

We study this map as described in the previous section. 1 d log b, restricted on ∂X+ , is the harmonic measure dω The differential 2πi ¯ of the domain C \ E with pole at infinity. The space Lp (∂X+ ), in a sense, is the Lp space with respect to the harmonic measure, but it should be mentioned that ∂X+ = (E − i0) ∪ (E + i0), i.e., an element f of Lp (∂X+ ) may have different values f (x + i0) and f (x − i0), x ∈ E, and H p (X+ ) is the closure of the set of holomorphic functions uniformly bounded on X+ with respect to this norm. ˜ ◦ π = λ we have the relation Since λ ˜b ◦ π = bd .

Thus, we get

f dω =

∂X+

˜+ ∂X

⎞ ⎛ 1⎝ ˜ d˜ f (Z)⎠ (Z) ω d ˜

(27)

π(Z)=Z

for every f ∈ L1 (∂X+ ). As it follows directly from (27), the covering (26) generates an isometric enclosure ˜ → H 2 (X+ ) v+ : H 2 (X) acting in the natural way

(v+ f )(Z) = f (π(Z)). Remark. As it was mentioned, the function b is not single valued but |b(z)| is a single valued function. We define the character µ ∈ Γ∗ by b(γ(z)) = µ(γ)b(z).


369

Let γj be the contour, that starts at infinity (or any other real point bigger than 1), go in the upper half-plane to the gap (aj , bj ) and then go back in the lower halfplane to the initial point. Assuming that b0 < · · · < aj < bj < aj+1 < · · · < a0 , we g+1−j have µ(γj ) = e−2πi g+1 , equivalently ω([bj , a0 ]) = g+1−j g+1 . Note that the system of the above contours γj is a generator of the free group Γ∗ (E). In other words a character α is uniquely defined by the vector α(γ1 ), α(γ2 ), . . . , α(γg ) ∈ Tg .

This sets a one-to-one correspondence between Γ∗ (E) and Tg . Recall the key role of the reproducing kernels k α in our construction. In this particular case they are especially well studied [10]. First of all, they have an analytic continuation (as multivalued functions) on the whole X, so we can write k α (Z). Theorem 6.2. For every α ∈ Γ∗ the reproducing kernel k α (Z) has on X exactly g simple poles that do not depend on α and g simple zeros. The divisor D+ = g j=1 Zj of zeros k α (Zj ) = 0 with the divisor of poles form the divisor div(k α ) = D+ − DC

(28)

that belongs to D(E), moreover (28) sets a one-to-one correspondence between D(E) and Γ∗ (E). The functions k α possess different representations, in particular, in terms of theta-functions [16], and the map D → α can be written explicitly in terms of Abelian integrals (the Abel map). Summary. The three objects J(E), D(E) and Γ∗ (E) are equivalent. Both maps Γ∗ (E) → D(E) and Γ∗ (E) → J(E) can be defined in terms of the reproducing kernels of the spaces H 2 (X+ , α), α ∈ Γ∗ (E). The first one is given by (28). It associates to k α (Z) the sets of its zeros and poles (the poles are fixed and the zeros vary with α). The matrix J(α) ∈ J(E) is defined as the matrix of the multiplication operator by z(Z) α α α α α α z(Z)eα s (Z) = ps es−1 (Z) + qs es (Z) + ps+1 es+1 (Z), Z ∈ X, s ∈ Z,

with respect to the basis −s

s αμ eα (Z). s (Z) = b (Z)K

(29)

It is really easy to see that J(α) is periodic: just recall that bg+1 is single valued, that is, µg+1 = 1, and therefore the spaces H 2 (X+ , α) and H 2 (X+ , αµ−(g+1) ) (and their reproducing kernels) coincide.

370


Now we can go back to the Renormalization Equation. Note that π acts on Γ(E) in the following natural way: ˜ πγ = {π(Z), Z ∈ γ} ∈ Γ(E), for γ ∈ Γ(E).

˜ → Γ∗ (E) is defined by duality: The map π ∗ : Γ∗ (E) (π ∗ α ˜ )(γ) = α(πγ). ˜

Theorem 6.3. Let T , T −1 : [−1, 1] → [−1, 1], be an expanding polynomial. Let J˜ ˜ ⊂ [−1, 1], and therefore there exists be a periodic Jacobi matrix with spectrum E ˜ such that ˜ such that E ˜ =U ˜ −1 [−1, 1] and a character α ˜ ∈ Γ∗ (E) a polynomial U J˜ = J(˜ α). Then ˜ J := J(π ∗ α) = JT (J) ˜ ◦ T , that is the periodic Jacobi matrix with spectrum E = U −1 [−1, 1], U := U satisfies the Renormalization Equation T ′ (z) ˜ −1 , V ∗ (z − J)−1 V = (T (z) − J) d where the isometry matrix V is defined by V |k := |kd . Remark 6.4. Let us mention that the Renormalization Equation can be rewritten equivalently in the form of polynomials equations, as it should be since we have ˜ in the other one, Jac(X). Equation the map from one algebraic variety, Jac(X), (6.3) is equivalent to, see [21], ˜ ∗, V ∗ T (J) = JV V∗

T (z) − T (J) V = T ′ (z)/d. z−J

Proof of Theorem 6.3. First we note, that for the operator multiplication by z(Z) ˜ in L2 (∂ X ˜ + ), the spectral paramin L2 (∂X+ ), the operator multiplication by z˜(Z) eter z0 and the isometry ˜ + ) → L2 (∂X+ ), (vf )(Z) = f (π(Z)), v : L2 (∂ X we have ∂X+

1 |(vf )(Z)|2 dω = z0 − z(Z)

It is evident, that

⎛

⎝1 d ˜+ ∂X

˜ π(Z)=Z

⎞ 1 ˜ 2 d˜ ⎠ |f (Z)| ω. z0 − z(Z)

(30)

1 T ′ (z0 )/d 1 = . d z0 − y T (z0 ) − x T (y)=x

Thus

˜ −1 . v ∗ (z0 − z(Z))−1 v = (T ′ (z0 )/d)(T (z0 ) − z˜(Z))

−n

˜μ ˜ ñ α˜ It remains to show that π transforms the basis vector e˜α n = b K ∗

eπndα

nd

=b K

∗

−nd

(π α)μ ˜

= (˜bn ◦ π)K π

∗

−n

(α˜ ˜μ

)

.

into


371

∗ ˜ Note that both Or, what is the same, that K α˜ ◦ π = K π α for all α ˜ ∈ Γ∗ (E). functions are of norm one in the same space H 2 (X+ , π ∗ α ˜ ), in particular, they have ˜ ∈ Γ∗ (E). Note, finally, that the divisor the same character of automorphy π ∗ α

˜ + ) − π −1 (D ˜ C ), div(k α˜ ◦ π) = π −1 (D

˜+ − D ˜ C , belongs to D(E), therefore k α˜ ◦ π is the reproducing where div(k α˜ ) = D kernel and the theorem is proved. Probably it would be better to call (6.3) the Renormalization Identity in the above theorem. The idea is that one can try to define J as the solution of (6.3) ˜ Indeed, in the case of one-sided matrices such an equation has with the given J. an unique solution. Now we demonstrate that in the two sided case for the given periodic J˜ we can find 2d−1 solutions. This proof is based on Theorem 6.2. For an alternative proof see [21]. To find all this solutions of (6.3) let us look a bit more carefully at the proof of Theorem 6.3. Note that the same identity (30) holds for any isometry v of the form vf = vθ f = θ(f ◦ π), where θ is a unimodular (|θ| = 1) function on ∂X+ . Concerning the second part of the proof, let us mention that the set of critical points of U splits into two sets: {cU } = T −1 {cU˜ } ∪ {cT }. Correspondingly,

(CU )j =

(CU )k,j +

k π(CU )k,j =(CU ˜ )k

(CT )j ,

˜ and the divisor of k α˜ ◦ π consists of two parts, the one that depends on α ˜ π −1 (D),

and the part that corresponds to the critical points of the polynomial T {(CT )j }d−1 j=1 , since ˜ + D = div(k α˜ ◦ π) = π −1 (D)

d−1 j=1

˜C) − (CT )j − π −1 (D {Zc,j }d−1 j=1

d−1 (CT )j . j=1

Thus we can fix an arbitrary system of points such that z(Zc,j ) belongs to the same gap in the spectrum E as the critical point (cT )j . If θ is the canonical product on X with the divisor div(θ) =

d−1 j=1

Zc,j −

d−1

(CT )j ,

j=1

372


˜ But to then θk α˜ ◦ π is the reproducing kernel simultaneously for all α ˜ ∈ Γ∗ (E). make θ unimodular (zeros and poles are symmetric) our choice is restricted just to Zc,j = (CT )j or Zc,j = τ (CT )j . Note that τ (CT )j − (CT )j is the divisor of the complex Green function b(cT )j . In this way we arrive at Theorem 6.5. For an expanding polynomial T , and a periodic Jacobi matrix J˜ = ˜ as in Theorem 6.3 there exist 2d−1 solutions of the RenormalJ(˜ α), α ˜ ∈ Γ∗ (E) ization Equation (6.3). Denote by µ(cT )j the character generated by the Green’s function b(cT )j , b(cT )j ◦ γ = µ(cT )j (γ)b(cT )j . Then these solutions are of the form ˜) , J := J (ηδ π ∗ α

ηδ :=

d−1

1

(1+δ(cT )j )

2 µ(c T )j

,

(31)

(Z)⎠ f (π(Z))

j=1

as before δ = {δ(cT )j }, Proof. We define the isometry

⎛

d−1

(vf )(Z) = ⎝

δ(cT )j = ±1.

1 2 (1+δ(cT )j )

b(cT )j

j=1

⎞

and then repeat the arguments of the proof of Theorem 6.3.

7. The Renormalization Equation for two-sided Jacobi matrices – general case In this section we assume that T (z) = z d − qdz d−1 + · · ·

is a monic expanding polynomial. Under this normalization T −1 : [−ξ, ξ] → [−ξ, ξ] for a certain ξ > 0. It was proved in [21] that the Renormalization Equation has 2d−1 solutions for every two sided Jacobi matrix J˜ with the spectrum in [−ξ, ξ], not only for periodic one. Moreover, they are the only possible solutions. For the reader’s convenience we formulate these theorems here. 2 By l± (s) we denote the spaces which are formed by {|s + k } with k ≤ 0 2 2 (s + 1). Correspondingly to these (s) ⊕ l+ and k ≥ 0 respectively, that is l2 (Z) = l− ˜ 2 (s). Recall that a (finite or infinite) decompositions we set J˜± (s) = Pl2± (s) J|l ± one-sided Jacobi matrix is uniquely determined by its so-called resolvent function r˜± (z, s) = s|(J˜± (s) − z)−1 |s .

(32)

We parametrize this set of solutions by a collections of vectors δ := {δc }c , where each component δc can be chosen as plus or minus one.

(33)


373

Theorem 7.1. Fix a vector δ of the form (33). For a given J˜ with the spectrum on [−ξ, ξ] define the Jacobi matrix J according to the following algorithm: For s ∈ Z we put 1 = −˜ r− (T (c), s), if δc = −1, (34) T (s) (c) and T (s) (c) = −˜ p2s+1 r˜+ (T (c), s + 1), if δc = 1, where the functions r˜± (z, s) are defined by (32). Then define the monic polynomial T (s) (z) of degree d by the interpolation formula T ′ (z) T (s) (z) = (z − q)T ′ (z)/d + T (s) (c). (z − c)T ′′ (c) ′ c:T (c)=0

Define the block

⎡

qsd

⎢psd+1 ⎢ ⎢ J (s) = ⎢ ⎢ ⎣

psd+1 qsd+1 .. .

psd+2 .. . psd+d−2

..

. qsd+d−2 psd+d−1

by its resolvent function

⎤

⎥ ⎥ ⎥ ⎥ ⎥ psd+d−1 ⎦ qsd+d−1

J T ′ (z)/d I 0 (z − J (s) )−1 0 = (s) , T (z)

where T (s) (z) is a monic polynomial of degree d. Finally define the entry psd+d by psd+1 . . . psd+d = p˜s+1 ˜ combined with such blocks and entries We claim that the matrix J = J(δ, J), over all s, satisfies (6.3). Theorem 7.2. [21] Theorem 7.1 describes the whole set of solutions of the Renormalization Equation. In [21] we concentrated only on one of the solutions of (6.3), namely that one that is related to the vector δ− = {−1, . . . , −1},

˜ = JT (J) ˜ for a periodic J. ˜ that is, all T (c) are defined by (34). Note, J(δ− , J) Precisely for this solution we proved (main Theorem 1.1 in [21]): (s)

Theorem 7.3. Let J˜ be a Jacobi matrix with the spectrum on [−ξ, ξ]. Then the ˜ with the spectrum on Renormalization Equation (6.3) has a solution J = J(δ− , J) T −1 ([−ξ, ξ]). Moreover, if mini |ti | ≥ 10ξ then J(δ− , J˜1 ) − J(δ− , J˜2 ) ≤ κJ˜1 − J˜2

with an absolute constant κ < 1.

(35)

374


Let us emphasize the special role of the position of the critical values: the ˜ is a contraction as soon as critical values are sufficiently far away transform JT (J) from the spectrum. In this work we add certain remarks concerning other solutions of (6.3). 7.1. The duality δ → −δ At least for one more solution of the Renormalization Equation is a contraction. Theorem 7.4. The dual solution of the Renormalization Equation J(J˜, −δ), pos˜ δ) the contractibility property. sesses simultaneously with J(J, It deals with the following universal involution acting on Jacobi matrices J → Jτ := Uτ JUτ ,

where Uτ |l = |1 − l .

(36)

Obviously V Uτ = Uτ S 1−d V . Thus, having J as a solution of the renormalization equation corresponding to J˜ we have simultaneously that S d−1 Jτ S 1−d solves the equation with the initial J˜τ . The following lemma describes which branch corresponds to which in this case. ˜ δ) then Lemma 7.5. Let J = J(J, S d−1 Jτ S 1−d = J(J˜τ , −δ).

(37)

Proof. We give a proof using the language of Section 5, so formally we prove the claim only for periodic matrices. Note that the involution (36) is strongly related to the standard involution τ (25) on X. Indeed, the function K(τ Z, α) has the divisor τ D+ − τ DC = (τ D+ − DC ) − (τ DC − DC ), that is, K(τ Z, α) =

K(Z, β) , bc1 (Z) . . . bcg (Z)

and β = να−1 , where ν = µc1 . . . µcg . Due to this remark and the property z(τ Z) = z(Z) we have (J(α))τ = J(νµα−1 ). (38) ˜ + ). Or, in Now we apply (38) to prove (37). Let J˜τ = J(˜ α) with α ˜ ∈ Γ∗ (X other words, J˜ = J(˜ µν˜α ˜ −1 ). Then by (31) J(J˜, δ) = J(ηδ π ∗ (˜ µν˜α ˜ −1 )),

ηδ :=

d−1

1

(1+δ(cT )j )

2 µ(c T )j

.

j=1

But π ∗ µ ˜ = µd and π ∗ (˜ ν ) = νηδ−1 (just to look at the characters of the correspond+ ing Blaschke products). Thus, having in mind that ηδ η−δ = ηδ+ , we obtain −1 ∗ J(J˜, δ) = J(µd νη−δ π (˜ α−1 )).


375

Using again (38) we get (J(J˜, δ))τ = J(µ1−d η−δ π ∗ (˜ α)) = S 1−d J(η−δ π ∗ (˜ α))S d−1 ,

and the lemma and Theorem 7.4 are proved.

Let Jul(T ) be the Julia set of a sufficiently hyperbolic polynomial T . Having two different contractive branches of solutions of the renormalization equation, following [14], to an arbitrary sequence ǫ = {ǫ0 , ǫ1 , ǫ2 , . . . },

ǫ j = δ± .

we can associate a limit periodic matrix J with the spectrum on Jul(T ). To this end we define J as the limit of Jn := J(ηǫ0 π ∗ ηǫ1 . . . π∗ ηǫn−1 )

(39)

and use Theorem 7.4. 7.2. Other solutions of the Renormalization Equation and the Ruelle operators Iterating the renormalization transform Jn+1 := J(Jn , δ− ), we obtain an almost periodic operator J = lim Jn with spectrum on the Julia set of the sufficiently expanding polynomial T (the key point, here, is of course property (35)). This operator possesses the following structure [21]: it is the direct sum of two (one-sided) Jacobi matrices J± . Moreover, the spectral measure of the operator J+ is the balanced measure µ on the Julia set. That is, it is the eigenmeasure of the Ruelle operator L∗ , where 1 (Lf )(x) = f (y). d T n(y)=x

The spectral measure ν of J− is the so-called Bowen–Ruelle measure. It is the eigenmeasure for L∗2 , f (y) (L2 f )(x) = , T ′ (x)2 T (y)=x

We conjecture that actually all branches of solutions of the renormalization equation are contractions for sufficiently hyperbolic T . At least the previous subsection looks as a quite strong indication in this direction: considering, instead of initial T , T 2 = T ◦ T or its bigger powers, we get, as in (39), several δ’s, ηδ = ηǫ0 π ∗ ηǫ1 . . . π∗ ηǫn−1 , possessing the contractibility property with respect to the polynomial T n and different from δ± (related to (π ∗ )n ). Similarly to the above statement we formulate Conjecture. Let T (z) be an expanding polynomial. For a given δ we factorize T ′ (z) = A1 (z)A2 (z) putting in the first factor all critical points related to δc = 1.

376


Denote by σ1,2 , the (nonnegative) eigenmeasures, corresponding to the Ruelle operators f (y) (LAi f )(x) = , (40) Ai (y)2 T (y)=x

L∗Ai σi

i.e., = ρi σi . Finally let J1,2 be the one-sided Jacobi matrices associated with σ1,2 . We conjecture that the iterations Jn+1 := J(Jn , δ), converges to the block matrix J = J− ⊕ J+ with J− = J1 and J+ = J2 . In particular this means that all such operators are limit periodic. In the support of this conjecture let us note that the statement holds true, say, for the polynomial of the form T 2 (z), and for A1 (z) = T ′ (z), A2 = T ′ (T (z)). 7.3. Shift transformations with the Lipschitz property We say that the direction η ∈ Γ∗ has the Lipschitz property with a constant C(η) if for all α, β ∈ Γ∗ J(ηα) − J(ηβ) ≤ C(η)J(α) − J(β). Then, one can get the contractibility of the map ηπ ∗ in two steps: ˜ ≤ C(η)J(π ∗ α ˜ ˜ ) − J(π ∗ β) ˜ ) − J(ηπ ∗ β) J(ηπ ∗ α ˜ ≤ C(η)κJ(˜ α) − J(β). Note, that in fact the situation is a bit more involved because we should be able to compare Jacobi matrices with different spectral sets, for example, when ˜1 = E ˜2 . But we just want to indicate the general idea. Note, in Ei = T −1 E˜i , E particular, that for directions ηδ of the form (31) such a comparison is possible. Of course, for our goal the constant C(η) should be uniformly bounded when we increase the level of hyperbolicity of T making κ smaller. However the key point of this remark (this way of proof) is that, actually, we do not need to constrain ourselves by the form of the vector η. Combining a “Lipschitz” shift by η (the direction is restricted just by this property) with a sufficiently contractive pull-back π ∗ we arrive at an iterative process that produces a limit periodic Jacobi matrix with the spectrum on the same Jul(T ). In the next subsection we give examples of directions with the required property, see Corollary 7.8. We do not have a proof of the Lipschitz property of ηδ ’s, but there is a good chance to generalize the result of the next subsection in a way that at least some of the directions ηδ will be also available. Finally, we would be very interested to know, whether there is in general a relation between the form of the “weight” vector η and the corresponding weights of the Ruelle operators (if any exists).


377

7.4. Quadratic polynomials and the Lipschitz property of the Darboux transform Consider the simplest special case T (z) = ρ(z 2 − 1) + 1, ρ > 2. Note that the ˜ is symmetric, moreover the matrix related to H 2 (π ∗ α ˜) spectral set E = T −1 E has zero main diagonal (as well as a one-sided matrix related to a symmetric ˜ which is very similar to measure). Now we introduce a decomposition of H 2 (π ∗ α) the standard decomposition into even and odd functions. ˜) We define the two-dimensional vector-function representation of f ∈ H 2 (π ∗ α $ $ # # ˜ ˜ 1 f (Z1 (Z)) g (Z) (41) → 1 ˜ , f → √ ˜ g2 (Z) 2 f (Z2 (Z)) where

#

$ $ # ˜ + f (Z2 (Z)) ˜ ˜ 1 f (Z1 (Z)) g1 (Z) , = ˜ − f (Z2 (Z)) ˜ ˜ g2 (Z) 2 f (Z1 (Z))

the first component, in a sense, is even and the second is odd. To be more precise, ˜+. let us describe the analytical properties of this object in ∂ X Note that due to 1 |f |2 dω = |f |2 d˜ ω 2 ˜ ∂ X+ ∂X+ ˜ π(Z)=Z

metrically it is from L2 with respect to ω ˜ , moreover the transformation is normpreserved. ˜+ , α ˜ ). Consider the second It is evident that the function g1 belongs to H 2 (X function. Note that the critical points of T are zero and infinity. For a small circle γ around the point T (0) = −ρ + 1 we have g2 ◦ γ = −g2 and the same property holds for a contour γ that surrounds infinity. Let us introduce ∆2 := ˜bT (0)˜b. Note that for the above contours we have ∆ ◦ γ = −∆. We are going to represent g2 in the form g2 = ∆ˆ g2 and to claim that gˆ2 has nice automorphic properties in ˜ + . Let us note that X ˜b z˜ − T (0) = ˜b2 z˜ − T (0) ˜bT (0) ∆2 ¯ \E ˜≃X ˜ + . So, the square root of this function is an outer function in the domain C is well defined. We put > ˜bφ := ˜b2 z˜ − T (0) (42) ρ∆2 and denote by η˜ the character generated by φ, φ ◦ γ = η(γ)φ. Thus (42) reduces the ramification of the function ∆ to thefunction φ, which is well defined in the domain, and to the elementary function z˜ − T (0).

378


Theorem 7.6. The transformation f → g1 ⊕ gˆ2 given by (41) is a unitary map from H 2 (π ∗ α ˜ ) to H 2 (˜ α) ⊕ H 2 (˜ αη˜). Moreover with respect to this representation # $# $ 0 φ¯ g1 zf → (43) φ 0 gˆ2 and

v+ f → f ⊕ 0, f ∈ H 2 (˜ α), where the isometry v+ : H (˜ α) → H 2 (π ∗ α ˜ ) is defined by (6). 2

Proof. By the definition of ∆ we have Further, since

g2 = ∆ˆ g2 , where gˆ2 ∈ H 2 (˜ αη˜). z1,2 = ±

>

z˜ − T (0) , ρ

we have, say for the second component, > 1 (zf )(Z(Z˜1 )) − (zf )(Z(Z˜2 )) z˜ − T (0) f (Z(Z˜1 )) + f (Z(Z˜2 )) = = φg1 . (44) ∆ 2 ρ∆2 2 Since on the boundary of the domain z˜ − T (0) = |φ|2 φ2 ∆2 = ρ ˜ + ) we have (the second expression is positive on ∂ X φ∆2 = φ

˜ on E.

(45)

Using this relation, similarly to (44), we prove the identity of the first components in (43). ˜ + ) → L2 (∂ X ˜ + ) with respect Theorem 7.7. The multiplication operator φ : L2 (∂ X to the basis systems (29) related to α ˜ and η˜α ˜ , respectively, is a two diagonal matrix Φ. Moreover, J(˜ α) − T (0) J(˜ ηα ˜ ) − T (0) Φ∗ Φ = , ΦΦ∗ = . (46) ρ ρ In other words, the transformation J(˜ α) → J(˜ ηα ˜ ) is the Darboux transform. Proof. First of all φ is a character-automorphic function with the character η˜ with a unique pole at infinity (˜bφ is an outer function). Therefore the multiplication operator acts from ˜bH 2 (˜ αµ ˜−1 ) to H 2 (˜ ηα ˜ ). Therefore, the operator Φ has only one non-trivial diagonal above the main diagonal. The adjoint operator has the symbol φ. According to (45) it has holomorphic continuation from the boundary inside the domain. Thus Φ∗ is a lower triangular matrix. Combining these two facts we get that Φ has only two non-trivial diagonals. Then, just comparing symbols of operators on the left and right parts of (46), we prove these identities.


379

Corollary 7.8. Let J˜1,2 be periodic Jacobi matrices with the spectrum on [−1, 1]. Let Drb(J˜1,2 , ρ) be their Darboux transforms. Then Drb(J˜1 , ρ) − Drb(J˜2 , ρ) ≤ C(ρ)J˜1 − J˜2 .

(47)

Proof. For the given J˜1,2 we define J1,2 via the quadratic polynomial T (z) = ρ(z 2 − 1) + 1. Being decomposed into even and odd indexed subspaces they are of the form $ # 0 Φ∗1,2 . J1,2 = Φ1,2 0 Due to the Theorem 7.3, that gives the uniform estimate for J1 − J2 , we have Φ1 − Φ2 ≤ κ(ρ)J˜1 − J˜2 , with κ(ρ) =

C ρ−2 ,

C is an absolute constant. Using (46) we get (47) with C(ρ) =

2ρC ρ−2 .

8. Double covering π(v) = τ v − 1/v 8.1. Over the simply-connected domain

The simplest expanding double coverings are T1 = v 2 −λ, λ > 2, and T2 = τ v−1/v, τ > 1. Denote by ξ1,2 the fixed points T1,2 (ξ1,2 ) = ξ1,2 , ξ1,2 > 0. To start with we give a complete description of finite difference operators related to these coverings over the simply-connected domain C \ E, where E = [−ξ1,2 , ξ1,2 ]. More precisely, we start with a Jacobi matrix J0 with constant coefficients. ξ −1 ). That is the Under the normalization σ(J0 ) = E we have J = 1,2 2 (S + S symbols of this operator (z∗ , b∗ ) are related by

ξ1,2 1 z∗ = (48) + b∗ , 2 b∗ (b∗ is the Green function of the domain C \ E). For π = T1 or π = T2 we consider the open Riemann surfaces Yc \F with Yc ≃ C, F = π −1 (E) and describe operators with the symbols (z ∗ , b∗ ): z ∗ = z∗ ◦ π

and (b∗ )2 = b∗ ◦ π.

(49)

The main difference between these two cases is that in the first one we have only one infinity on Yc (∞ ∈ C) and in the second case there are two infinities: 0, ∞ ∈ C. Correspondingly an intrinsic basis contains the reproducing kernels related only to one given point in the first case and to two specific points in the second case. As result the multiplication operator by v with respect to this basis is a Jacobi matrix in the first case and a five diagonal matrix (of a special form, see Lemma 4.1) in the second.

380


8.1.1. π = v 2 − λ. Due to (48), (49) we have

1 ξ1 ∗ 2 . + (b ) v2 − λ = 2 (b∗ )2

(50)

Since v is the symbol of a Jacobi matrix,

v ∼ SΛ1 + Λ0 + Λ1 S −1 ,

where Λ0,1 are diagonal of period two matrices, and (b∗ )2 ∼ S 2 , we have from (50) (1) (1)

λ0 λ1 = ξ1 /2, (0)

(0)

(51)

λ0 + λ1 = 0, (0) (λ0 )2

+

(1) (λ0 )2

+

(1) (λ1 )2

= λ.

8.1.2. π = τ v − 1/v. Repeating arguments (49), (48) we get instead of (50)

1 ξ2 + (b∗ )2 . τ v − 1/v = ∗ 2 2 (b ) or

1 ξ2 τ v2 − 1 = (52) + (b∗ )2 v. ∗ 2 2 (b ) In this case we have a five diagonal matrix, v ∼ S 2 Λ2 + SΛ1 + Λ0 + Λ1 S −1 + Λ2 S −2 ,

but of the specific structure: depending of ordering of infinities one of coefficients (2) (2) (2) (2) λ0 , λ1 vanishes. Say, λ1 = 0, respectively λ0 = 0. Using (b∗ )2 ∼ S 2 we get from (52) τ S 2 Λ2 S 2 Λ2 =

ξ2 4 S Λ2 , 2

ξ2 3 S Λ1 , 2 ξ2 τ (S 2 Λ2 Λ0 + SΛ1 SΛ1 + Λ0 S 2 Λ2 ) = S 2 Λ0 , 2

τ (S 2 Λ2 SΛ1 + SΛ1 S 2 Λ2 ) =

(53)

ξ2 2 S Λ1 S −1 , 2 ξ2 τ (S 2 Λ22 S −2 + SΛ21 S −1 + Λ20 + Λ21 + Λ22 ) − I = (S 2 Λ2 S −2 + Λ2 ). 2

τ (S 2 Λ2 Λ1 S −1 + SΛ1 Λ0 + Λ0 SΛ1 + Λ1 SΛ2 ) =

(2)

That is τ λ0 = ξ2 /2 and the second relation in (53) is an identity. Further, ξ (0) ξ (0) (1) (1) τ λ0 λ1 = − λ0 = λ1 (54) 2 2 and the fourth relation is an identity. Finally, from the last equation we get (0)

(1)

(1)

(λ0 )2 + (λ0 )2 + (λ1 )2 = 1/τ. Thus (54), (55) are counterparts of (51) in this case.

(55)


381

8.2. The Renormalization Equation in the general case For π : C → C given by T2 (that is Xc ≃ C and Yc ≃ C) let E be a system of intervals, a subset of [−ξ2 , ξ2 ]. As usual F = π −1 (E). For a system of points on Xc \ E P1 = ∞, P2 , . . . , Pl , (56) we define

bl∗ = bP1 . . . bPl .

We consider z∗ given by the identical map Xc operator J˜ related to this symbol, to a character α system of infinities (56), is of the form ⎡ ∗ ∗ ... ∗ ⎢ ∗ ∗ ... ∗ ⎢ ⎢ ∗ ... ∗ ⎢ ⎢ . .. ⎢ . .. ⎢ ⎢ ˜ ∗ ∗ ∗ ... J =⎢ ⎢ ∗ ∗ ... ⎢ ⎢ ∗ ... ⎢ ⎢ .. ⎣ .

because, actually, z∗ (Pj ) = ∞ only for j = 1. The system of infinities on Yc is given by (1)

where

(j) (j) {Q1 , Q2 }

(1)

(l)

→ C. The finite difference ∈ Γ∗X and to the ordering ⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥, ∗ ⎥ ⎥ ∗ ⎥ ⎥ .. ⎥ . ⎦ ∗ (l)

Q1 = ∞, Q2 = 0, . . . , Q1 , Q2 , =π

−1

(57)

(Pj ), j = 1, . . . , l. Respectively,

1 z = z∗ ◦ π = τ v − , v ∗

(b∗ )2 = b∗ ◦ π.

We are interested to find a relation between the initial operator J˜ and the finite difference operator J with the symbol v related to the character π ∗ α ∈ Γ∗Y and to the ordering system of infinities (57). Let us point out that the symbol of J is v, not z ∗ . It gives us an opportunity to iterate the procedure, which appears to be a certain rescaling. Let us prove the following lemma. Lemma 8.1. Let T (z) =

Q(z) P (z) .

Define

(Lf )(x) =

1 f (y). d T (y)=x

Then L

1 1 P ′ (z) 1 T ′ (z) = + . z−x d P (z) d T (z) − x

382


Proof. Note that

Q(y)−xP (y)=0

1 Q′ (z) − xP ′ (z) = z−y Q(z) − xP (z)

and then collect corresponding terms Q′ (z) − Q(z)P ′ (z)/P (z) + {Q(z) − xP (z)}P ′ (z)/P (z) Q′ (z) − xP ′ (z) = Q(z) − xP (z) Q(z) − xP (z) 1 P ′ (z) Q′ (z)P (z) − Q(z)P ′ (z) + . = P (z) P 2 (z) T (z) − x

Theorem 8.2. The following relation holds V∗ (z − J)−1 V =

1 P ′ (z) 1 ′ ˜ −1 , + T (z)(T (z) − J) d P (z) d

(58)

where J˜ = z∗ (α) and J = v(π ∗ α). Proof. We use the previous lemma and the functional representation of all operators involved in (58). 8.3. A vector representation of H 2 (π ∗ α) Due to 1 |f |2 ω ∗ = E 2 F

π(Q)=P

|f |2 ω∗

we have a certain representation of f ∈ H 2 (π ∗ α) $ # $ # 1 f (Q1 (P )) g (P ) → 1 , f → √ g2 (P ) 2 f (Q2 (P ))

where

#

(59)

$ $ # 1 f (Q1 (P )) + f (Q2 (P )) g1 (P ) = , g2 (P ) 2 f (Q1 (P )) − f (Q2 (P )) as a two-dimensional vector-function. Let us describe analytical properties of this object (metrically it is from L2 with respect to ω∗ , moreover the transformation is norm-preserved). It is evident that the function g1 belongs to H 2 (α). Consider the second function. Let c± be critical points i π ′ (c± ) = 0 ⇒ c± = ± √ . τ √ For a small circle γ around the point π(c± ) = ±2i τ we have g2 ◦ γ = −g2 . Let us introduce ∆2 := bπ(c+ ) bπ(c− ) . Note that ∆ ◦ γ = −∆. Further, since the ratio bπ(c+ ) z − π(c+ ) (z) = eiC bπ(c− ) z − π(c− )


383

is single-valued in Xc \ E both Green functions have the same character of automorphy, which we denote by ν ∈ Γ∗X . Thus, we get g2 = ∆˜ g2 ,

where

g˜2 ∈ H 2 (αν −1 ).

(60)

We can summarize the result of this subsection as Theorem 8.3. The transformation f → g1 ⊕ g2 given by (59) is a unitary map from H 2 (π ∗ α) to H 2 (α) ⊕ ∆H 2 (αν −1 ). Moreover, V f → f ⊕ 0,

f ∈ H 2 (α),

where the isometry V : H 2 (α) → H 2 (π ∗ α) is defined by (21). Also, ∗

π α kQ = kPα ⊕ (±∆∆(P )kPαν ±

for the reproducing kernel

π∗ α kQ ±

2

−1

)

∗

∈ H (π α), π(Q± ) = P .

Extending this vector representation onto L2 -spaces we get immediately Theorem 8.4. The transformation f → g1 ⊕ g˜2 given by (59) and (60) is a unitary map from L2 (π ∗ α) to L2 (α) ⊕ L2 (αν −1 ). With respect to this representation the multiplication operator by z ∗ = z∗ ◦ π is of the form $ # 0 z (α) . (61) z∗ (π ∗ α) → ∗ −1 0 z∗ (αν ) To get a similar representation for the multiplication operator by v we need to introduce the following notations. Let us note that z 2 + 4τ ∆2 is an outer function in the domain C \ E. So, the square root of this function is well defined. We put 8 z 2 + 4τ . b∞ φ := b2∞ ∆2 Since on the boundary of the domain b2∞

φ2 ∆2 = z 2 + 4τ = |φ|2

(the second expression is positive on E) we have φ∆2 = φ

on E.

(62)

Lemma 8.5. The multiplication operator φ : L2 (α) → L2 (αν −1 ) with respect to the bases systems related to the infinities {P1 , . . . , Pl } has as many diagonals as z∗ (α) and z∗ (αν −1 ). Moreover, φz∗ (α) = z∗ (αν −1 )φ, φ∗ φ = z2∗ (α) + 4τ,

φφ∗ = z2∗ (αν −1 ) + 4τ.

(63)

384


Proof. First of all φ is a character-automorphic function with the character ν −1 , therefore the multiplication operator acts from L2 (α) to L2 (αν −1 ). Since b∞ φ is an outer function, φ has a unique pole at infinity, and, therefore, the operator φ has the same structure over diagonal as the operator multiplication by z. The adjoint operator has the symbol φ. According to (62) it has analytic continuation from the boundary inside the domain with the only pole at infinity. Thus φ∗ is also of the same structure over the main diagonal as z∗ (α) or z∗ (αν −1 ). Combining these two facts we get that the whole structure of φ coincides with the structure of the matrix z∗ (α). Then, just comparing symbols of operators on the left and right parts of (63), we prove these identities. Theorem 8.6. With respect to the decomposition L2 (π ∗ α) ≃ L2 (α) ⊕ L2 (αν −1 ) the multiplication operator by v is of the form $ # 1 z∗ (α) φ∗ . (64) v(π ∗ α) ≃ −1 φ z∗ (αν ) 2τ Let us mention that according to (63), the operators given in (61) and (64) commute and satisfy the identity, which is generated by the symbols identity z ∗ = τ v − 1/v. 8.4. One sided matrices and the expanding transform π(v) = τ v − τ −1 v Note that in this normalization v = 1 is the positive fixed point, π(1) = 1. Put E0 = [−1, 1]. For a continuous function f on E1 = π −1 ([−1, 1]) = [−1, −1 + we define (Lf )(x) =

1 1 ] ∪ [1 − , 1] τ τ

1 f (v). 2 π(v)=x

The conjugate operator acts on measures

L∗ : C(E0 )∗ → C(E1 )∗ . Let f0 , f1 , f2 , . . . be a certain orthonormal system with respect to a (positive) measure ν ∈ C(E0 )∗ then f0 ◦ π, f1 ◦ π, f2 ◦ π, . . .

is orthonormal system with respect to µ := L∗ ν. Note that if the first system form basis in L2dν the second one form basis in the set of “even” functions from L2dμ , the functions that are invariant with respect to the substitution f (− ττ−1 v ) = f (v). Example. Let f0 , f1 , f2 , . . . be orthonormal polynomials in L2dν . f0 ◦ π, f1 ◦ π, f2 ◦ π, . . . is a certain orthonormal system in L2dμ consisting of “polynomials” of v and 1/v, similarly to the systems that generate CMV matrices: 1, v, 1/v, v 2 , 1/v 2 . . . .


385

Making a small modification in this procedure, we orthogonalize

2

2 τ −1 τ −1 τ −1 τ −1 , τv − , (τ v)2 − 1, τ v + , (τ v)2 + ... v v v v and denote the orthonormal system by e0 , e1 , e2 , . . . . It is evident that e2k = fk ◦ π

and

τ −1 gk (π(v)), τv + v where gk is also orthonormal system of polynomials but with respect to the measure (x2 + 4τ (τ − 1))dν(x), since

2 τ −1 τ −1 τv + . = x2 + 4τ (τ − 1), f or x = τ v − v v e2k+1 (v) =

Let J be the Jacobi matrix, corresponding to the multiplication operator in L2dν with respect to the basis of the orthonormal polynomials. The given J we want to describe the multiplication operator in L2dμ with respect to {ek }. We decompose L2dμ onto even and odd subspaces. Then # $ τ −1 J 0 → , τv − 0 J∗ v where J∗ is the Jacobi matrix corresponding to the measure (x2 + 4τ (τ − 1))dν(x). Further, $ # τ −1 0 Φ∗ . → τv + Φ 0 v It is quite evident that Φ is an upper triangular matrix. We get that # $ 1 J Φ∗ v → , 2τ Φ J∗ and # $ 1 J −Φ∗ −1/v → . 2(τ − 1) −Φ J∗

Therefore #

J 2 − Φ∗ Φ ΦJ − J∗ Φ

−JΦ∗ + Φ∗ J∗ J∗ 2 − ΦΦ∗

$

=

#

−4τ (τ − 1) 0 0 −4τ (τ − 1)

Thus Φ can be found in the upper-lower triangular factorization Φ∗ Φ = J 2 + 4τ (τ − 1),

and for J∗ we have J∗ = ΦJΦ−1 .

$

.

386

F. Peherstorfer and P. Yuditskii Thus for

⎡

we have

⎡

⎢ ⎢ ⎢ ⎢ 1 ⎢ ⎢ π ∗ (J) = ⎢ 2τ ⎢ ⎢ ⎢ ⎢ ⎣

0 ⎢ J = ⎣ p1 0 λ0 p1 0 0 0 0

0 0 p1 λλ01 0 0 p1 p2 λ0

p1 0 .. .

0 λ1 p2 0 0 .. .

p2 .. .

0 0 p2 λλ12 0

..

⎤ .

⎥ ⎦

0 λ2 p3

⎤

0 0 ..

.

0 .. .

The matrix is self-adjoint and λn are defined by the recursion λ2n = 4τ (τ − 1) + p2n+1 + p2n −

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎦

p2n p2n−1 . λ2n−2

with the initial data λ20 = p21 + 4τ (τ − 1),

λ21 = p21 + p22 + 4τ (τ − 1).

Theorem 8.7. Let ν be the spectral measure of A, dν(x) = 0|(A − z)−1 |0 . (65) x−z Then π ∗ (A) is a self-adjoint operator with the cyclic vector |0 and the spectral measure µ = L∗ ν. Proof. By Definition 1.5 and (65) we have G# $−1 # $ # $H |0 |0 A − 2τ z Φ∗ ∗ −1 , 0|(π (A) − z) |0 =2τ Φ A∗ − 2τ z 0 0 =2τ 0|(A − 2τ z − Φ∗ (A∗ − 2τ z)−1 Φ)−1 |0

=2τ 0|(A − 2τ z − Φ∗ Φ(A − 2τ z)−1 )−1 |0

Since

L

1 v−z

=2τ 0|(A − 2τ z − (A2 + 4τ (1 − τ ))(A − 2τ z)−1 )−1 |0 2τ dν(x) = x − 2τ z − (x2 + 4τ (1 − τ ))(x − 2τ z)−1 (x − 2τ z)dν(x) . = 2τ z 2 − 2xz − 2(1 − τ )

(x) =

1 2

τ v− τ −1 v =x

x − 2τ z 1 = , v−z 2τ z 2 − 2xz − 2(1 − τ )

Finite Difference Operators with a Finite-band Spectrum we get 0|(π ∗ (A) − z)−1 |0 = and the theorem is proved.

1 1 (x)dν(x) = d(L∗ ν)(v) L v−z v−z

387

Using Ruelle’s Theorem with respect to the map π(v) we can summarize our considerations by the following theorem. Theorem 8.8. The iterative procedure An+1 = π ∗ (An ) converges to the operator A = limn→∞ An with the spectral measure µ which is the eigenmeasure for the Ruelle operator L∗ µ = µ. The operator A is the multiplication operator by the independent variable in L2dμ with respect to the following basis e2k (v) = ek (π(v)),

k τ −1 k e2k+1 (v) = τ v + cj ej (π(v)), v j=0

e0 (v) = 1,

where the coefficients ckj with ckk > 0 are uniquely determined due to the orthogonality condition e2k+1 , el = δ2k+1,l , l ≤ 2k + 1. Moreover, ek (v) is a rational function of v such that ek (A)|0 = |k , and ⎤ ⎡ 0 c0 c10 c20 . . . ⎢ 0 c11 c21 . . . ⎥ −1 ⎦=Φ . ⎣ .. .. .. . . .

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[7] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, 1987. [8] D. Damanik, R. Killip, and B. Simon Perturbation of orthogonal polynomials with periodic recursion coefficients, preprint. [9] B.A. Dubrovin, I.M. Krichever and S.P. Novikov, Dynamical systems. IV, 173–280, Springer-Verlag, Berlin, 1990. [10] J. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, New York/Berlin, 1973. [11] L. Faybusovich and M. Gekhtman, On Schur flows, J. Phys. A 32 (1999), no. 25, 4671–4680. [12] M. Hasumi, Hardy Classes on Infinitely Connected Riemann Surfaces, Lecture Notes in Math. 1027, Springer-Verlag, Berlin and New York, 1983. [13] J. Herndon, Limit periodicity of sequences defined by certain recurrence relations; and Julia sets, Ph.D. thesis, Georgia Institute of Technology, 1985. [14] O. Knill, Isospectral deformations of random Jacobi operators. Comm. Math. Phys. 151 (1993), no. 2, 403–426. [15] G. MacLane, Concerning the uniformization of certain Riemann surfaces allied to the inverse-cosine and inverse-gamma surfaces, Trans. AMS, 62 (1947) 99–113. [16] D. Mumford, Lectures on theta-functions. (Russian) Translated from the English by D.Yu. Manin. Translation edited and with a preface by Yu.I. Manin. With appendices by Khirosi Umemura [Hiroshi Umemura] and Takakhiro Shiota [Takahiro Shiota]. “Mir”, Moscow, 1988. 448 pp. [17] P. van Moerbeke, The spectrum of Jacobi matrices. Invent. Math. 37 (1976), no. 1, 45–81. [18] P. van Moerbekke, D. Mumford, The Spectrum of Difference Operators and Algebraic Curves, Acta Math. 143 (1979), no. 1-2, 93–154. [19] L. Pastur and A. Figotin Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Berlin, 1986. [20] F. Peherstorfer and P. Yuditskii, Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces, J. Approx. Theory 139 (2006), 91–106 [21] F. Peherstorfer, A. Volberg and P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and singular continuous spectrum, Math. Res. Lett. 13 (2006), 215–230. [22] F. Pehersorfer, A. Volberg, P. Yuditskii, Two weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials, submitted. [23] M. Shapiro, V. Vinnikov, P. Yuditskii, Finite Difference Operators with a Finite-band Spectrum, Mat. Fiz. Anal. Geom. 11 (2004), no. 3, 331–340. [24] B. Simon, Orthogonal Polynomials on the Unit Circle, AMS Colloquium Series, American Mathematical Society, Providence, RI, expected 2004. [25] M. Sodin, P. Yuditski, The limit-periodic finite-difference operator on l2 (Z) associated with iterations of quadratic polynomials. J. Statist. Phys. 60 (1990), no. 5-6, 863–873. [26] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of characterautomorphic functions, J. Geom. Anal. 7 (1997), no. 3, 387–435.


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[27] G. Teschel, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, 72, American Mathematical Society, Providence, RI, 2000. [28] E.B. Vinberg, Real entire functions with prescribed critical values, Problems of group theory and homological algebra, Yaroslavl. Gos. U., Yaroslavl, 1989, 127–138. [29] V. Vinnikov and P. Yuditskii, Functional models for almost periodic Jacobi matrices and the Toda hierarchy, MAG 9(2002), 206–219. F. Peherstorfer and P. Yuditskii Abteilung f¨ ur Dynamische Systeme und Approximationstheorie Universit¨ at Linz A-4040 Linz, Austria e-mail: [email protected] e-mail: [email protected]


Connections Between Dynamical Systems and Crossed Products of Banach Algebras by Z Christian Svensson, Sergei Silvestrov and Marcel de Jeu Abstract. Starting with a complex commutative semi-simple regular Banach algebra A and an automorphism σ of A, we form the crossed product of A by the integers, where the latter act on A via iterations of σ. The automorphism induces a topological dynamical system on the character space Δ(A) of A in a natural way. We prove an equivalence between the property that every nonzero ideal in the crossed product has non-zero intersection with the subalgebra A, maximal commutativity of A in the crossed product, and density of the non-periodic points of the induced system on the character space. We also prove that every non-trivial ideal in the crossed product always intersects the commutant of A non-trivially. Furthermore, under the assumption that A is unital and such that Δ(A) consists of infinitely many points, we show equivalence between simplicity of the crossed product and minimality of the induced system, and between primeness of the crossed product and topological transitivity of the system. Mathematics Subject Classification (2000). Primary 47L65 Secondary 16S35, 37B05, 54H20. Keywords. Crossed product; Banach algebra; ideals, dynamical system; maximal abelian subalgebra.

1. Introduction A lot of work has been done in the direction of connections between certain topological dynamical systems and crossed product C ∗ -algebras. In [6] and [7], for example, one starts with a homeomorphism σ of a compact Hausdorff space X and constructs the crossed product C ∗ -algebra C(X) ⋊α Z, where C(X) is the algebra of continuous complex valued functions on X and α is the automorphism of C(X) naturally induced by σ. One of many results obtained is equivalence between simplicity of the algebra and minimality of the system, provided that X consists of infinitely many points, see [1], [3], [6], [7] or, for a more general result, [8]. In [5],

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a purely algebraic variant of the crossed product is considered, with more general classes of algebras than merely continuous functions on compact Hausdorff spaces serving as “coefficient algebras”. For example, it was proved there that, for such crossed products, the analogue of the equivalence between density of non-periodic points of a dynamical system and maximal commutativity of the “coefficient algebra” in the associated crossed product C ∗ -algebra is true for significantly larger classes of coefficient algebras and associated dynamical systems. In this paper, we go beyond these results and investigate the ideal structure of some of the crossed products considered in [5]. More specifically, we consider crossed products of complex commutative semi-simple regular Banach algebras A with the integers under an automorphism σ : A → A. In Section 2 we give the most general definition of the kind of crossed product that we will use throughout this paper. We also mention the elementary result that the commutant of the coefficient algebra is automatically a commutative subalgebra of the crossed product. The more specific setup that we will be working in is introduced in Section 3. There we also introduce some notation and mention two basic results concerning a canonical isomorphism between certain crossed products, and an explicit description of the commutant of the coefficient algebra in one of them. According to [7, Theorem 5.4], the following three properties are equivalent: • The non-periodic points of (X, σ) are dense in X; • Every non-zero closed ideal I of the crossed product C ∗ -algebra C(X) ⋊α Z is such that I ∩ C(X) = {0}; • C(X) is a maximal abelian C ∗ -subalgebra of C(X) ⋊α Z.

In Section 4, an analogue of this result is proved for our setup. A reader familiar with the theory of crossed product C ∗ -algebras will easily recognize that if one chooses A = C(X) for X a compact Hausdorff space in Corollary 4.5 below, then the crossed product is canonically isomorphic to a norm-dense subalgebra of the crossed product C ∗ -algebra coming from the considered induced dynamical system. We also combine this with a theorem from [5] to conclude a stronger result for the Banach algebra L1 (G), where G is a locally compact abelian group with connected dual group. In Section 5, we prove the equivalence between algebraic simplicity of the crossed product and minimality of the induced dynamical system in the case when A is unital with its character space consisting of infinitely many points. This is analogous to [7, Theorem 5.3], [1, Theorem VIII 3.9], the main result in [3] and, as a special case of a more general result, [8, Corollary 8.22] for the crossed product C ∗ -algebra. In Section 6, the fact that the commutant of A always has non-zero intersection with any non-zero ideal of the crossed product is shown. This should be compared with the fact that A itself may well have zero intersection with such ideals, as Corollary 4.5 shows. The analogue of this result in the context of crossed product C ∗ -algebras appears to be open.

Connections Between Dynamical Systems and Crossed Products. . .

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Finally, in Section 7 we show equivalence between primeness of the crossed product and topological transitivity of the induced system, in the case when A is unital and has an infinite character space. The analogue of this in the context of crossed product C ∗ -algebras is [7, Theorem 5.5].

2. Definition and a basic result Let A be an associative commutative complex algebra and let Ψ : A → A be an algebra automorphism. Consider the set A ⋊Ψ Z = {f : Z → A | f (n) = 0 except for a finite number of n}.

We endow it with the structure of an associative complex algebra by defining scalar multiplication and addition as the usual pointwise operations. Multiplication is defined by twisted convolution, ∗, as follows; (f ∗ g)(n) = f (k) · Ψk (g(n − k)), k∈Z

where Ψk denotes the k-fold composition of Ψ with itself. It is trivially verified that A ⋊σ Z is an associative C-algebra under these operations. We call it the crossed product of A by Z under Ψ. A useful way of working with A⋊Ψ Z is to write elements f, g ∈ A⋊Ψ Z in the form f = n∈Z fn δ n , g = m∈Z gm δ m , where fn = f (n), gm = g(m), addition and scalar multiplication are canonically defined, and multiplication is determined by (fn δ n ) ∗ (gm δ m ) = fn · Ψn (gm )δ n+m , where n, m ∈ Z and fn , gm ∈ A are arbitrary. Clearly one may canonically view A as an abelian subalgebra of A ⋊Ψ Z, namely as {f0 δ 0 | f0 ∈ A}. The following elementary result is proved in [5, Proposition 2.1]. Proposition 2.1. The commutant A′ of A in A ⋊Ψ Z is abelian, and thus it is the unique maximal abelian subalgebra containing A.

3. Setup and two basic results In what follows, we shall focus on cases when A is a commutative complex Banach algebra, and freely make use of the basic theory for such A, see, e.g., [2]. As conventions tend to differ slightly in the literature, however, we mention that we call a commutative Banach algebra A semi-simple if the Gelfand transform on A is injective, and that we call it regular if, for every subset F of the character space ∆(A) that is closed in the Gelfand topology and for every φ0 ∈ ∆(A) \ F , there exists an a ∈ A such that * a(φ) = 0 for all φ ∈ F and * a(φ0 ) = 0. All topological considerations of the character space ∆(A) will be done with respect to its Gelfand topology (the weakest topology making all elements in the image of the Gelfand transform of A continuous on ∆(A)).

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Now let A be a complex commutative semi-simple regular Banach algebra, and let σ : A → A be an algebra automorphism. As in [5], σ induces a map σ ) : ∆(A) → ∆(A) (where ∆(A) denotes the character space of A) defined by σ )(µ) = µ ◦ σ −1 , µ ∈ ∆(A), which is automatically a homeomorphism when ∆(A) is endowed with the Gelfand topology. Hence we obtain a topological dynamical * → A * (where A * system (∆(A), σ )). In turn, σ ) induces an automorphism σ * : A denotes the algebra of Gelfand transforms of all elements of A) defined by σ * (* a) = K Therefore we can form the crossed product A⋊ * σ* Z. We also mention * a ◦) σ −1 = σ(a). that when speaking of ideals, we will always mean two-sided ideals. In what follows, we shall make frequent use of the following fact. Its proof consists of a trivial direct verification. Theorem 3.1. Let A be a commutative semi-simple Banach algebra and σ be an *→ A * as above. Then the map automorphism, inducing an automorphism σ * : A n n * ⋊σ* Z defined by a δ → < Φ : A ⋊σ Z → A n n δ is an isomorphism of n∈Z n∈Z a * algebras mapping A onto A. Before stating the next result, we make the following basic definitions.

Definition 3.2. For any nonzero n ∈ Z we set

)n (µ)}. Pern (∆(A)) = {µ ∈ ∆(A) | µ = σ

Furthermore, we denote the non-periodic points by @ (∆(A)\ Pern (∆(A)). Per∞ (∆(A)) = n∈Z\{0}

* put Finally, for f ∈ A,

supp(f ) = {µ ∈ ∆(A) | f (µ) = 0}. ′

* ⋊σ* Z: * in A Theorem 3.3. We have the following explicit description of A * and for all n ∈ Z : supp(fn ) ⊆ Pern (∆(A))}. *′ = { fn δ n | fn ∈ A, A n∈Z

* trivially separates the points of Proof. This follows from [5, Corollary 3.4], as A ∆(A) and Pern (∆(A)) is a closed set.

4. Three equivalent properties In this section we shall conclude that, for certain A, two different algebraic properties of A ⋊σ Z are equivalent to density of the non-periodic points of the naturally associated dynamical system on the character space ∆(A), and hence obtain equivalence of three different properties. The analogue of this result in the context of crossed product C ∗ -algebras is [7, Theorem 5.4]. We shall also combine this with a theorem from [5] to conclude a stronger result for the Banach algebra L1 (G), where G is a locally compact abelian group with connected dual group.


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In [5, Theorem 4.8], the following result is proved. Theorem 4.1. Let A be a complex commutative regular semi-simple Banach algebra, σ : A → A an automorphism and σ ) the homeomorphism of ∆(A) in the Gelfand topology induced by σ as described above. Then the non-periodic points are dense * is a maximal abelian subalgebra of A * ⋊σ* Z. In particular, in ∆(A) if and only if A A is maximal abelian in A ⋊σ Z if and only if the non-periodic points are dense in ∆(A). We shall soon prove another algebraic property of the crossed product equivalent to density of the non-periodic points of the induced system on the character space. First, however, we need two easy topological lemmas. Lemma 4.2. Let x ∈ ∆(A) be such that the points σ )i (x) are distinct for all i such that −m ≤ i ≤ n, where n and m are positive integers. Then there exist an open set Ux containing x such that the sets σ )i (Ux ) are pairwise disjoint for all i such that −m ≤ i ≤ n.

Proof. It is easily checked that any finite set of points in a Hausdorff space can be separated by pairwise disjoint open sets. Separate the points σ )i (x) with disjoint open sets Vi . Then it is readily verified that the set Ux := σ )m (V−m ) ∩ σ )m−1 (V−m+1 ) ∩ · · · ∩ V0 ∩ σ )−1 (V1 ) ∩ · · · ∩ σ )−n (Vn )

is an open neighbourhood of x with the required property.

Lemma 4.3. The non-periodic points of (∆(A), σ )) are dense if and only if the set Pern (∆(A)) has empty interior for all positive integers n.

Proof. Clearly, if there is a positive integer n0 such that Pern0 (∆(A)) has nonempty interior, the non-periodic points are not dense. For the converse, we recall that ∆(A) is a Baire space since it is locally compact and Hausdorff, and note that we may write L Pern (∆(A)). ∆(A)\ Per∞ (∆(A)) = n>0

If the set of non-periodic points is not dense, its complement has non-empty interior, and as the sets Pern (∆(A)) are clearly all closed, there must exist an integer n0 > 0 such that Pern0 (∆(A)) has non-empty interior since ∆(A) is a Baire space. We are now ready to prove the promised result.

Theorem 4.4. Let A be a complex commutative semi-simple regular Banach algebra, σ : A → A an automorphism and σ ) the homeomorphism of ∆(A) in the Gelfand topology induced by σ as described above. Then the non-periodic points are dense in ∆(A) if and only if every non-zero ideal I ⊆ A ⋊σ Z is such that I ∩ A = {0}.

* ⋊σ* Z. Proof. We first assume that Per∞ (∆(A)) = ∆(A), and work initially in A n * ⋊σ* Z is a non-zero ideal, and that f = f δ ∈ I. By Assume that I ⊆ A n∈Z n definition, only finitely many fn are non-zero. Denote the set of integers n for

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which fn ≡ 0 by S = {n1 , . . . , nr }. Pick a non-periodic point x ∈ ∆(A) such that fn1 (x) = 0 (by density of Per∞ (∆(A)) such x exists). Using the fact that x is not periodic we may, by Lemma 4.2, choose an open neighbourhood Ux of x such that σ )−ni (Ux ) ∩ σ )−nj (Ux ) = ∅ for ni = nj , ni , nj ∈ S. Now by regularity of A we can * that is non-zero in σ find a function g ∈ A )−n1 (x), and vanishes outside σ )−n1 (Ux ). )−n )δ n . This is an element in I and clearly the Consider f ∗ g = n∈Z fn · (g ◦ σ coefficient of δ n1 is the only one that does not vanish on the open set Ux . Again * that is non-zero in x and vanishes outside by regularity of A, there is an h ∈ A )−n1 )fn1 ]δ n1 is a non-zero monomial belonging Ux . Clearly h ∗ f ∗ g = [h · (g ◦ σ to I. Now any ideal that contains a non-zero monomial automatically contains a * Namely, if ai δ i ∈ I then [ai δ i ] ∗ [(ai ◦ σ * By non-zero element of A. )i )δ −i ] = a2i ∈ A. the canonical isomorphism in Theorem 3.1, the result holds for A ⋊σ Z as well. * σ* Z. For the converse, assume that Per∞ (∆(A)) = ∆(A). Again we work in A⋊ It follows from Lemma 4.3 that since Per∞ (∆(A)) = ∆(A), there exists an integer n > 0 such that Pern (∆(A)) has non-empty interior. As A is assumed to be * such that supp(f ) ⊆ Pern (∆(A)). Consider now the regular, there exists f ∈ A ideal I = (f + f δ n ). Using that f vanishes outside Pern (∆(A)), we have )−i )δ i ] ∗ [f δ j + f δ n+j ] ai δ i (f + f δ n )aj δ j = [ai · (aj ◦ σ

= [ai · (aj ◦ σ )−i ) · (f ◦ σ )−i )]δ i+j + [ai · (aj ◦ σ )−i ) · (f ◦ σ )−i )]δ i+j+n . Therefore any element in I may be written in the form i (bi δ i + bi δ n+i ). As i runs only through a finite subset of Z, this is not a non-zero monomial. In particular, * Hence I intersects A * trivially. By the canonical it is not a non-zero element in A. isomorphism in Theorem 3.1, the result carries over to A ⋊σ Z. Combining Theorem 4.1 and Theorem 4.4, we now have the following result.

Corollary 4.5. Let A be a complex commutative semi-simple regular Banach algebra, σ : A → A an automorphism and σ ) the homeomorphism of ∆(A) in the Gelfand topology induced by σ as described above. Then the following three properties are equivalent: )) are dense in ∆(A); • The non-periodic points Per∞ (∆(A)) of (∆(A), σ • Every non-zero ideal I ⊆ A ⋊σ Z is such that I ∩ A = {0}; • A is a maximal abelian subalgebra of A ⋊σ Z. We shall make use of Corollary 4.5 to conclude a result for a more specific class of Banach algebras. We start by recalling a number of standard results from the theory of Fourier analysis on groups, and refer to [2] and [4] for details. Let G be a locally compact abelian group. Recall that L1 (G) consists of equivalence classes of complex valued Borel measurable functions of G that are integrable with respect to a Haar measure on G, and that L1 (G) equipped with convolution product is a commutative regular semi-simple Banach algebra. A group homomorphism γ : G → T from a locally compact abelian group to the unit circle is called a character of G. The set of all continuous characters of G forms a group Γ, the


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dual group of G, if the group operation is defined by If γ ∈ Γ and if

(γ1 + γ2 )(x) = γ1 (x)γ2 (x) (x ∈ G; γ1 , γ2 ∈ Γ). f*(γ) =

f (x)γ(−x)dx

G

(f ∈ L1 (G)),

then the map f → f*(γ) is a non-zero complex homomorphism of L1 (G). Conversely, every non-zero complex homomorphism of L1 (G) is obtained in this way, and distinct characters induce distinct homomorpisms. Thus we may identify Γ with ∆(L1 (G)). The function f* : Γ → C defined as above is called the Fourier transform of f ∈ L1 (G), and is precisely the Gelfand transform of f . We denote the set of all such f* by A(Γ). Furthermore, Γ is a locally compact abelian group in the Gelfand topology. In [5, Theorem 4.16], the following result is proved. Theorem 4.6. Let G be a locally compact abelian group with connected dual group and let σ : L1 (G) → L1 (G) be an automorphism. Then L1 (G) is maximal abelian in L1 (G) ⋊σ Z if and only if σ is not of finite order. Combining Corollary 4.5 and Theorem 4.6 the following result is immediate. Corollary 4.7. Let G be a locally compact abelian group with connected dual group and let σ : L1 (G) → L1 (G) be an automorphism. Then the following three statements are equivalent. • σ is not of finite order; • Every non-zero ideal I ⊆ L1 (G) ⋊σ Z is such that I ∩ L1 (G) = {0}; • L1 (G) is a maximal abelian subalgebra of L1 (G) ⋊σ Z.

5. Minimality versus simplicity Recall that a topological dynamical system is said to be minimal if all of its orbits are dense, and that an algebra is called simple if it lacks non-trivial proper ideals. Theorem 5.1. Let A be a complex commutative semi-simple regular unital Banach algebra such that ∆(A) consists of infinitely many points, and let σ : A → A be an algebra automorphism of A. Then A ⋊σ Z is simple if and only if the naturally induced system (∆(A), σ )) is minimal.

Proof. Suppose first that the system is minimal, and assume that I is a proper ideal of A ⋊σ Z. Note that I ∩ A is a proper σ- and σ −1 -invariant ideal of A. By basic theory of Banach algebras, I ∩ A is contained in a maximal ideal of A (note that I ∩ A = A as A is unital and I was assumed to be proper), which is the kernel of an element µ ∈ ∆(A). Now I ∩ A is a σ * - and σ *−1 -invariant proper * all of whose elements vanish in µ. Invariance of this ideal non-trivial ideal of A, implies that all of its elements even annihilate the whole orbit of µ under σ ). But by

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minimality, every such orbit is dense and hence I ∩ A = {0}. By semi-simplicity of A, this means I ∩ A = {0}, so I = {0} by Corollary 4.5. For the converse, assume that there is an element µ ∈ ∆(A) whose orbit O(µ) is not dense. By * that vanishes on O(µ). Then clearly the regularity of A there is a nonzero g ∈ A * ideal generated by g in A ⋊σ* Z consists of finite sums of elements of the form (fn δ n ) ∗ g ∗ (hm δ m ) = [fn · (g ◦ σ )−n ) · (hm ◦ σ )−n )]δ n+m , and hence the coefficient of every power of δ in this ideal must vanish in µ, whence the ideal is proper. Hence by Theorem 3.1, A ⋊σ Z is not simple.

6. Every non-zero ideal has non-zero intersection with A′ We shall now show that any non-zero ideal in A ⋊σ Z has non-zero intersection with A′ . This should be compared with Corollary 4.5, which says that a non-zero ideal may well intersect A solely in 0. We have no information on the validity of the analogue of this result in the context of crossed product C ∗ -algebras. Theorem 6.1. Let A be a complex commutative semi-simple regular Banach algebra, and σ : A → A an automorphism. Then every non-zero ideal I in A ⋊σ Z has nonzero intersection with the commutant A′ of A in A ⋊Ψ Z, that is I ∩ A′ = {0}. * ⋊σ* Z. When Per∞ (∆(A)) = ∆(A), the result follows Proof. As usual, we work in A immediately from Corollary 4.5. We will use induction on the number of nonzero terms in an element f = n∈Z fn δ n to show that it generates an ideal that * ′ non-trivially. The starting point for the induction, namely when intersects A n f = fn δ with non-zero fn , is clear since any such element generates an ideal * non-trivially, as was shown in the proof of Theorem 4.4. that even intersects A Now assume inductively that the conclusion of the theorem is true for the ideals * σ* Z with r non-zero terms for some positive integer generated by any element of A⋊ r, and consider an element f = fn1 δ n1 + · · · + fnr+1 δ nr+1 . By multiplying from the right with a suitable we obtain an element in the ideal generated by mr element gi δ i such that g0 ≡ 0. If some of the other gi are zero we f of the form g = i=0 are done by induction hypothesis, so we may assume this is not the case. We may * since otherwise we are of course also assume that g is not in the commutant of A also done. This means, by Theorem 3.3, that there is such j that 0 < j ≤ mr and supp(gj ) ⊆ Perj (∆(A)). Pick an x ∈ supp(gj ) such that x = σ )−j (x) and gj (x) = 0. As ∆(A) is Hausdorff we can choose an open neighbourhood Ux of * x such that Ux ∩ σ )−j (Ux ) = ∅. Regularity of A implies existence of an h ∈ A such that h ◦ σ )−j (x) = 1 and h vanishes identically outside of σ )−j (Ux ). Now r * such σ −i )δ i . Using regularity of A again we pick a function a ∈ A g∗h = m i=0 gi ·(h◦) r −i i a·g ·(h◦ σ ) )δ , that a(x) = 1 and a vanishes outside Ux . We have a∗ g ∗ h = m i i=0 which is in the ideal generated by f . Now a·g0 ·h is identically zero since a·h = 0. On the other hand, a·gj ·(h◦) σ −j ) is non-zero in the point x. Hence a∗g∗h is a non-zero


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element in the ideal generated by f whose number of non-zero coefficient functions is less than or equal to r. By the induction hypothesis, such an element generates * non-trivially. By Theorem 3.1 it follows an ideal that intersects the commutant of A that every non-zero ideal in A ⋊σ Z intersects A′ non-trivially.

7. Primeness versus topological transitivity We shall show that for certain A, A ⋊σ Z is prime if and only if the induced system (∆(A), σ ) ) is topologically transitive. The analogue of this result in the context of crossed product C ∗ -algebras is in [7, Theorem 5.5].

Definition 7.1. The system (∆(A), σ )) is called topologically transitive if for any pair of non-empty open sets U, V of ∆(A), there exists an integer n such that σ )n (U ) ∩ V = ∅. Definition 7.2. The algebra A ⋊σ Z is called prime if the intersection between any two non-zero ideals I, J is non-zero, that is I ∩ J = {0}. For convenience, we also make the following definition. n * ⋊σ* Z → A * is defined by E( Definition 7.3. The map E : A n∈Z fn δ ) = f0 .

To prove the main theorem of this section, we need the two following topological lemmas. Lemma 7.4. If (∆(A), σ )) is not topologically transitive, then there exist two disjoint invariant non-empty open sets O1 and O2 such that O1 ∪ O2 = ∆(A).

Proof. As the system is not topologically transitive, there exist non-empty open sets U, V ⊆ ∆(A) such )n (U ) ∩ V = ∅. Now thatn for any integer n we have σ ) (U ) is an invariant non-empty open set. Then O1 clearly the set O1 = n∈Z σ is an invariant closed set. It follows that O2 = ∆(A)\O1 is an invariant open set containing V . Thus we even have that O1 ∪O2 = ∆(A), and the result follows.

Lemma 7.5. If (∆(A), σ )) is topologically transitive and there is an n0 > 0 such that ∆(A) = Pern0 (∆(A)), then ∆(A) consists of a single orbit and is thus finite.

Proof. Assume two points x, y ∈ ∆(A) are not in the same orbit. As ∆(A) is Hausdorff we may separate the points x, σ(x), . . . , σn0 −1 (x), y by pairwise disjoint open sets V0 , V1 , . . . , Vn0 −1 , Vy . Now consider the set Ux := V0 ∩ σ )−1 (V1 ) ∩ σ )−2 (V2 ) ∩ · · · ∩ σ )−n0 +1 (Vn0 −1 ).

n0 −1 i 0 −1 i Clearly the sets Ax = ∪ni=0 σ ) (Ux ) and Ay = ∪i=0 σ ) (Vy ) are disjoint invariant non-empty open sets, which leads us to a contradiction. Hence ∆(A) consists of one single orbit under σ ).

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C. Svensson, S. Silvestrov and M. de Jeu We are now ready for a proof of the following result.

Theorem 7.6. Let A be a complex commutative semi-simple regular unital Banach algebra such that ∆(A) consists of infinitely many points, and let σ be an automorphism of A. Then A ⋊σ Z is prime if and only if the associated system (∆(A), σ )) on the character space is topologically transitive. Proof. Suppose (∆(A), σ )) is not topologically transitive. Then there exists, by Lemma 7.4, two disjoint invariant non-empty open sets O1 and O2 such that * ⋊σ* Z by k(O1 ) (the O1 ∪ O2 = ∆(A). Let I1 and I2 be the ideals generated in A * set of all functions in A that vanish on O1 ) and k(O2 ) respectively. We have that E(I1 ∩ I2 ) ⊆ E(I1 ) ∩ E(I2 ) = k(O1 ) ∩ k(O2 ) = k(O1 ∪ O2 ) = k(∆(A)) = {0}.

* σ* Z is an ideal and E(I) = {0}, then I = {0}. It is not difficult to see that if I ⊆ A⋊ Namely, suppose F = n fn δ n ∈ I and fi = 0 for some integer i. Since A is unital, * and thus δ −1 ∈ A * ⋊σ* Z. So F ∗ δ −i ∈ I and hence E(F ∗ δ −i ) = fi = 0 so is A * ⋊σ* Z is not prime. which is a contradiction, so I = {0}. Hence I1 ∩ I2 = {0} and A By Theorem 3.1, neither is A ⋊σ Z. Next suppose that (∆(A), σ ) ) is topologically transitive. Assume that Per∞ (∆(A)) is not dense. Then by Lemma 4.3 there is an integer n0 > 0 such that Pern0 (∆(A)) has non-empty interior. As Pern0 (∆(A)) is invariant and closed, topological transitivity implies that ∆(A) = Pern0 (∆(A)). This, however, is impossible since by Lemma 7.5 it would force ∆(A) to consist of a single orbit and hence be finite. Thus Per∞ (∆(A)) is dense after all. Now let I and J be two non-zero proper ideals in A ⋊σ Z. Unitality of A assures us that I ∩A and J ∩A are proper invariant ideals of A and density of Per∞ (∆(A)) assures us that they are non-zero, by Theorem 4.5. Consider AI = {µ ∈ ∆(A) | µ(a) = 0 for all a ∈ I ∩ A} and AJ = {ν ∈ ∆(A) | ν(b) = 0 for all b ∈ J ∩ A}. Now by Banach algebra theory a proper ideal in a commutative unital Banach algebra A is contained in a maximal ideal, and a maximal ideal of A is always precisely the set of zeroes of some ξ ∈ ∆(A). This implies that both AI and AJ are non-empty, and semi-simplicity of A assures us that they are proper subsets of ∆(A). Clearly they are also closed and invariant under σ ) and σ )−1 . Hence ∆(A) \ AI and ∆(A) \ AJ are invariant non-empty open sets. By topological transitivity we must have that these two sets intersect, hence that AI ∪ AJ = ∆(A). This means that there exists η ∈ ∆(A) and a ∈ I ∩ A, b ∈ J ∩ A such that η(a) = 0, η(b) = 0 and hence that η(ab) = 0. Hence 0 = ab ∈ I ∩ J, and we conclude that A ⋊σ Z is prime. Acknowledgments This work was supported by a visitor’s grant of the Netherlands Organisation for Scientific Research (NWO), The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Crafoord Foundation, The Royal Physiographic Society in Lund, The Royal Swedish Academy of Sciences and The Swedish Research Council. ¨ We are also grateful to Jun Tomiyama and Johan Oinert for useful discussions.


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References [1] Davidson, K.R., C ∗ -algebras by example, American Mathematical Society, Rhode Island, 1996. [2] Larsen, R., Banach algebras: an introduction, Marcel Dekker, Inc., New York, 1973. [3] Power, S.C., Simplicity of C ∗ -algebras of minimal dynamical systems, J. London Math. Soc. 18, 534–538, 1978. [4] Rudin, W., Fourier analysis on groups, Interscience Publishers, New York-London, 1962. [5] Svensson, C., Silvestrov, S., de Jeu, M., Dynamical Systems and Commutants in Crossed Products, Internat. J. Math., Vol. 18 (2007), no. 4, 455–471. [6] Tomiyama, J., Invitation to the C ∗ -algebras and topological dynamics, World Sci., Singapore New Jersey Hong Kong, 1987. [7] Tomiyama, J., The interplay between topological dynamics and theory of C ∗ -algebras, Lecture Note No.2, Global Anal. Research Center, Seoul, 1992. [8] Williams, D.P., Crossed Products of C ∗ -algebras, Mathematical Surveys and Monographs, American Mathematical Society, Province, R.I., 2007. Christian Svensson Mathematical Institute Leiden University P.O. Box 9512 2300 RA Leiden, The Netherlands and Centre for Mathematical Sciences Lund University Box 118 SE-221 00 Lund, Sweden e-mail: [email protected] Sergei Silvestrov Centre for Mathematical Sciences Lund University Box 118 SE-221 00 Lund, Sweden e-mail: [email protected] Marcel de Jeu Mathematical Institute Leiden University P.O. Box 9512 2300 RA Leiden, The Netherlands e-mail: [email protected]


Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the Area Fran¸coise Truc

Abstract. This paper is devoted to the asymptotics of eigenvalues for a Schrödinger operator Hh = −h2 Δ+V on L2 (Rm ), in the case when the potential V does not fulfill the non degeneracy condition: V (x) → +∞ as |x| → +∞. For such a model, the point is that the set defined in the phase space by: Hh ≤ λ may have an infinite volume, so that the Weyl formula which gives the behaviour of the counting function has to be revisited. We recall various results in this area, in the classical context (h = 1 and λ → +∞), as well as in the semi-classical one (h → 0) and comment the different methods. In Sections 3, 4 we present our joint works with A. Morame (Université de Nantes, e-mail: [email protected]),where we consider a degenerate potential V (x) = f (y)g(z), where g is assumed to be a homogeneous positive function of m variables , smooth outside 0, and f is a smooth and strictly positive function of n variables, with a minimum in 0. In the case where f (y) → +∞ as |y| → +∞, the operator has a compact resolvent and we give the asymptotic behaviour, for small values of h, of the number of eigenvalues less than a fixed energy. Then, without assumptions on the limit of f , we give a sharp estimate of the low eigenvalues, using a Born Oppenheimer approximation. With a refined approach we localize also higher energies. In the case when the degree of homogeneity is not less than 2, we can even assume that the order of these energies is like the inverse power of the square of h. Finally we apply the previous methods to a class of potentials in Rd , d ≥ 2, which vanish on a regular hypersurface. Mathematics Subject Classification (2000). Primary 35P20. Keywords. Eigenvalues, semi-classical asymptotics, minimax principle, BornOppenheimer approximation, Weyl formula, degenerate potentials.

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1. Introduction Let V be a nonnegative, real and continuous potential on Rm , and h a parameter in ]0, 1]. The spectral asymptotics of the operator Hh = −h2 ∆ + V on L2 (Rm ) have been intensively studied. More precisely it is well known [16] that Hh is essentially selfadjoint with compact resolvent when V (x) → +∞ as |x| → +∞ (we shall say that V is non degenerate). Moreover, denoting by N (λ, Hh ) the number of eigenvalues less than a fixed energy λ, the following semiclassical asymptotics hold, as h → 0: m/2 (λ − V (x))+ dx . (1.1) N (λ, Hh ) ∼ h−m (2π)−m vm Rm

In this formula, vm denotes the volume of the unit ball in Rm , and the notation W+ means the positive value of W . Let us note that the classical asymptotics are also given by the formula (1.1), provided we let h = 1 and λ → +∞. In both cases, the result points out the asymptotic correspondence between the number of eigenstates with energy less than λ and the volume in phase space of the set {(x, ξ), f (x, ξ) ≤ λ}, where f (x, ξ) = ξ 2 + V (x) is the principal symbol of Hh . In this paper we propose a review of results concerning the degenerate case: the potential V does not tend to infinity with |x|, so that the volume in phase space of the previous set may be infinite.

2. The Tauberian approach Let us explain how the problem of the degenerate case came from the non degenerate one. In 1950 De Wet and Mandl ([3]) proved the formula (1.1) in its classical version, provided V (x) ≥ 1 and two more conditions on V : 1) a smoothness condition: V differentiable and |∇V | = ◦(V )

m/2 2) a Tauberian type condition: let Φ(V, λ) = Rm (λ− V (x))+ dx; it is assumed ′ ′ that there exists c and c such that: cΦ(V, λ) ≤ λΦ (V, λ) ≤ c′ Φ(V, λ). The first condition is local and the second is global. This last condition was needed to use a Tauberian technique, which consists on studying the asymptotic behavior of the Green’s function of the operator H1 and applying a Tauberian theorem. Refinements were done by Titchmarsh, Levitan and Kostjucenko,([22], [11], [12]) and then Rosenbljum ([18]) proved that the formula (1.1) holds with “maximal” weakening conditions on V : 1) the smoothness condition is replaced by a condition on the “L1 -modulus of continuity” on unit cubes and by the following assumption: V (y) ≤ C ′ V (x) if |x − y| ≤ 1. 2) the Tauberian type condition becomes: σ(2λ, V ) ≤ Cσ(λ, V ) (for large λ), where σ(λ, V ) denotes the volume of the set {x ∈ Rm ; V (x) < λ}.

Born-Oppenheimer-type Approximations

405

Solomyak ([21]) makes the following remark: Lemma 2.1. Let V be a positive a-homogeneous potential: V (x) ≥ 0; V (tx) = ta V (x) for any t ≥ 0 (a > 0). If moreover V (x) is strictly positive (V (x) = 0 if x = 0) the spectrum of H1 is discrete and the formula (1.1) takes the form: 2m+am (V (x))−m/a dx N (λ, H1 ) ∼ γm,a λ 2a S m−1

(γm,a is a constant depending only on the parameters m and a). From that lemma comes out naturally the idea of investigating the spectrum without the condition of strict positivity (and thus in a case of degeneracy of the potential); the two main results are ([21]]: Theorem 2.2. The formula of Lemma 2.1 still holds for a positive a-homogeneous potential such that J(V ) = S m−1 (V (x))−m/a dx is finite.

The second result deals with a case where J(V ) is infinite: let V (x) = F (y, z), y ∈ Rn , z ∈ Rp , n + p = m, m ≥ 2, such that F (sy, tz) = sb ta−b F (y, z) (with 0 < a < b) and F (y, z) > 0 for |z||y| = 0. Denote by λj (y) the eigenvalues of the 2b , then: operator −∆z + F (y, z) in L2 (Rp ) and let s = 2+a−b Theorem 2.3. 2m+am m n > N (λ, H1 ) ∼ γn,s λ 2b If Σ(λj (y))−n/s dx b a S m−1 2m+am n a(a + 2) m if F (y, z)−m/a dx. = N (λ, H1 ) ∼ γm,a λ 2b ln λ b a 2b(a − b) S n−1 S p−1 The proof is based on variational techniques and spectral estimates proved in ([18]). But on a heuristic level the result can be understood in the framework of the theory of Schr¨ odinger operators with operator potential. This last approach can be found in ([17]) where D. Robert extended the theory of pseudodifferential operators in the form developed by H¨ ormander to pseudodifferential operators with operator symbols. It was thus possible to study cases where the operator has a compact resolvent but the condition lim∞ V (x) = +∞ is not fulfilled. As an example it gives the asymptotics of N (λ, H1 ) for the 2-dimensional potential V (y, z) = y 2k (1 + z 2 )l , where k et l are strictly positive. The asymptotics are the following: Theorem 2.4. If k > l

N (λ, H1 ) ∼ γ1 λ

if k = l

N (λ, H1 ) ∼ γ2 λ

if k < l

N (λ, H1 ) ∼ γ3 λ

l+k+1 2l 2k+1 2k 2k+1 2k

ln λ .

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The constants γi depend only on k and l, but the first one γ1 takes in account the trace of the operator (−∆z + z 2k )−(k+1)/2l in L2 (R). In the 2-dimensional case let us mention the results of B. Simon ([19]). He first recalls Weyl’s famous result: let H be the Dirichlet Laplacian in a bounded region Ω in R2 , then the following asymptotics hold: 1 N (λ, H) ∼ λ|Ω| 2 and then he considers special regions Ω for which the volume (denoted by |Ω|) is infinite but the spectrum of the Laplacian is still discrete. These regions are of the type: Ωμ = {(y, z); |y||z|μ ≤ 1}. Actually the problem can be derived from the study of the asymptotics of Schr¨ odinger operators with the homogeneous potential: V (y, z) = |y|α |z|β . In order to get these “non-Weyl” asymptotics, he uses the Feynman-Kac formula and the Karamata-Tauberian theorem, but the main tool is what he calls “sliced bread inequalities”, which can be seen as a kind of Born-Oppenheimer approximation. More precisely let H = −∆ + V (y, z) be defined on Rn+p , and denote by λj (y) the eigenvalues of the operator −∆z + V (y, z) in L2 (Rp ). (If the z’s are electron coordinates and the y’s are nuclear coordinates, the λj (y) are the Born-Oppenheimer curves.) He proves the following lemma: Tre−tH ≤ Σj e−t(−Δy +λj (y))

(when the second term exists). Thus he gets the two following coupled results:

Theorem 2.5. If H = −∆ + |y|α |z|β and α < β, then N (λ, H) ∼ cν λ

2ν+1 2

β+2 ). 2α 1

Corollary: if H = −∆Ωμ (µ > 1), then

N (λ, H) ∼ cμ λ 2μ+1 .

Theorem 2.6. If H = −∆ + |y|α |z|α , then Corollary: if H = −∆Ωμ (µ = 1), then

(ν =

N (λ, H) ∼

N (λ, H) ∼

1 1 1+ α πλ

ln λ.

1 π λ ln λ.

The constant cμ depends only on µ, and the constant cμ takes in account the trace of the operator (−∆z + |z|β )−ν in L2 (R).

3. The min-max approach The result presented in this section is based on the method of Courant and Hilbert, the min-max variational principle. It turns out that this method can be applied to operators in L2 (Rm ) with principal symbols which can degenerate on some non bounded manifold of T ∗ (Rm ). It is the case for the Schrödinger operator with a magnetic field H = (Dx − A(x))2 , which degenerates on {(x, ξ) ∈ T ∗ (Rm ); ξ = A(x)}. If the magnetic field B = dA fulfills the so-called magnetic bottle conditions


407

(mainly: lim∞ B(x) = ∞) the spectrum is discrete ([1]) and the classical asymptotics were established by Colin de Verdière ([2]) using the min-max method. The semiclassical version of the result is given in ([23]). In ([14]), the min-max method is performed to get semiclassical asymptotics for a large class of degenerate potentials, namely potentials of the following form: x = (y, z) ∈ Rn × Rp , n + p = m, m ≥ 2 p

V (x) = f (y)g(z), f ∈ C(Rn ; R∗+ ),

g ∈ C(R ; R+ ), g(tz) = ta g(z) (a > 0) ∀t > 0 , g(z) > 0 ∀z = 0. 2

(3.1)

p

The spectrum of the operator −∆z + g(z) in L (R ) is discrete and positive. Let us denote by µj its eigenvalues. It is easy to make the following remark: Remark 3.1. If f (y) → +∞ as |y| → +∞ then Hh = −h2 ∆ + V has a compact resolvent. Of course if f was supposed to be homogeneous, the asymptotics would be given by Theorem 2.3. Here the assumption on f is only a locally uniform regularity: ∃ b, c > 0 s.t. c−1 ≤ f (y) and |f (y) − f (y ′ )| ≤ cf (y)|y − y ′ |b ,

for any y, y ′ verifying |y − y ′ | ≤ 1.

Theorem 3.2. Let us assume the previous conditions on f and g. Then there exists σ, τ ∈]0, 1[ such that, for any λ > 0, one can find h0 ∈]0, 1[, C1 , C2 > 0 in order to have (1 − hσ C1 )nh,f (λ − hτ C2 ) ≤ N (λ; Hh ) ≤ (1 + hσ C1 )nh,f (λ + hτ C2 ) ∀h ∈]0, h0 [ n/2 if nh,f (λ) = h−n (2π)−n vn Σj∈N [λ − h2a/(2+a) f 2/(2+a) (y)µj ]+ dy . Rn

Provided some additional conditions on f , the previous result can be refined as follows: Theorem 3.3. If moreover one can find a constant C3 such that, for any µ > 1: f −p/a (y)dy ≤ C3 f −p/a (y)dy , {y,f (y) 0 such that 1 k |y| ≤ f (y) ≤ C|y|k for |y| > 1, C then if k > a

N (λ, Hh ) ≈ h−m

if k = a

N (λ, Hh ) ≈ h−m ln

if k < a

1 h pa N (λ, Hh ) ≈ h−n− k .

4. Born-Oppenheimer-type estimates In last section we have investigated the asymptotic behavior of the number of * h = −h2 ∆ + f (y)g(z). eigenvalues less then λ of H * h look like. This Theorem 3.2 gives us a hint of what should eigenvalues of H can be done using Born-Oppenheimer-type methods. We assume as in last section that: g ∈ C ∞ (Rm \ {0}) is homogeneous of degree a > 0, and assume the following for f : f ∈ C ∞ (Rn ), ∀α ∈ Nn , (|f (y)| + 1)−1 ∂yα f (y) ∈ L∞ (Rn ) 0 < f (0) = infn f (y) y∈R

f (0) < lim inf f (y) = f (∞)

(4.1)

|y|→∞

∂ 2 f (0) > 0. ∂ 2 f (0) denotes the hessian matrix in 0. 4.1. Using homogeneity * h by f (0), we can change the parameter h and assume that By dividing H f (0) = 1 .

Let us define: ℏ = h (3.1) to get:

2/(2+a)

(4.2)

and change z in zℏ; we can use the homogeneity of g

* ℏ) , * h ) = ℏa sp (H (4.3) sp (H 2 ℏ 2 2 * with H = ℏ Dy + Dz + f (y)g(z) . Let us denote as usually the increasing sequence of eigenvalues of Dz2 + g(z), (on L2 (Rm )), by (µj )j>0 . The associated eigenfunctions will be denoted by (ϕj )j : By homogeneity (3.1) the eigenvalues of Qy (z, Dz ) = Dz2 + f (y)g(z), on L2 (Rm )), for a fixed y, are given by the sequence (λj (y))j>0 , where : λj (y) = µj f 2/(2+a) (y).

Born-Oppenheimer-type Approximations So as in [15] we get: *ℏ ≥ H

+

ℏ2 Dy2 + µ1 f 2/(2+a) (y)

,

.

409

(4.4)

This estimate is sharp as we will see below. Then using the same kind of estimate as (4.4), one can see that * ℏ ) ≥ µ1 f 2/(2+a) (∞) . inf spess (H

(4.5)

We are in the Born-Oppenheimer approximation situation described by A. Martinez in [13]: the “effective ” potential is given by λ1 (y) = µ1 f 2/(2+a) (y), the first eigenvalue of Qy , and the assumptions on f ensure that this potential admits one unique and nondegenerate well U = {0}, with minimal value equal to µ1 . Hence we can apply Theorem 4.1 of [13] and get: Theorem 4.1. Under the above assumptions, for any arbitrary C > 0, there exists * ℏ ) admits a finite number of h0 > 0 such that, if 0 < ℏ < h0 , the operator (H eigenvalues Ek (ℏ) in [µ1 , µ1 + Cℏ], equal to the number of the eigenvalues ek of μ1 < ∂ 2 f (0)y, y > in [0, +C] such that: Dy2 + 2+a . * ℏ ) = λk ℏ2 D2 + µ1 f 2/(2+a) (y) + O(ℏ2 ) . Ek (ℏ) = λk (H (4.6) y * ℏ ) has an asymptotic expansion More precisely Ek (ℏ) = λk (H Ek (ℏ) ∼ µ1 + ℏ ( ek + αkj ℏj/2 ).

(4.7)

If Ek (ℏ) is asymptotically non degenerate, then there exists a quasimode φℏk (y, z) ∼ ℏ−mk e−ψ(y)/ℏ ℏj/2 akj (y, z) ,

(4.8)

j≥1

j≥0

satisfying

C0−1 ≤ ℏ−mk e−ψ(y)/ℏ ak0 (y, z) ≤ C0

ℏ−mk e−ψ(y)/ℏ akj (y, z) ≤ Cj ;

; ; ; j/2 ; * ℏ − µ1 − ℏek − ; H α ℏ kj ; ;

ℏ−mk e−ψ(x)/ℏ

0≤j≤J

(4.9)

1≤j≤J

ℏj/2 akj (x, y) ≤ CJ ℏ(J+1)/2 .

The formula (4.7) implies

* ℏ ) = µ1 + ℏλk Dy2 + µ1 < ∂ 2 f (0) y , y > + O(ℏ3/2 ) , λk (H 2+a

(4.10)

and when k = 1, one can improve O(ℏ3/2 ) into O(ℏ2 ). The function ψ is defined by: ψ(y) = d(y, 0), where d denotes the Agmon distance related to the degenerate metric µ1 f 2/(2+a) (y)dy 2 .

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4.2. Improving Born-Oppenheimer methods * ℏ . Let us make the change of We are interested now with the lower energies of H variables (y, z) → (y, f 1/(2+a) (y)z) . (4.11) The Jacobian of this diffeomorphism is f m/(2+a) (y), so we perform the change of test functions: u → f −m/(4+2a) (y)u, to get a unitary transformation. Thus we get that * ℏ ) = sp (H ) ℏ) sp (H (4.12) ℏ 2 n ) where H is the self-adjoint operator on L (R × Rm ) given by ) ℏ = ℏ2 D2 + f 2/(2+a) (y) D2 + g(z) H y z 2 + ℏ2 (∇f (y)Dy )(zDz ) (2 + a)f (y) (4.13) m 1 |∇f (y)|2 − f (y)∆f (y) [(zDz ) − i ] + iℏ2 (2 + a)f 2 (y) 2 1 m2 + ℏ2 |∇f (y)|2 [(zDz )2 + ]. (2 + a)2 f 2 (y) 4 The only significant role up to order 2 in ℏwill be played actually by the first ) ℏ = ℏ2 D2 + f 2/(2+a) (y) D2 + g(z) . operator, namely: H 1 y z This leads to:

Theorem 4.2. . Under the assumptions (3.1) and (4.1), for any fixed integer N > 0, there exists a positive constant h0 (N ) verifying: for any ℏ ∈]0, h0 (N )[, for any k ≤ N and any j ≤ N such that µj < µ1 f 2/(2+a) (∞) ,

* ℏ ) such that there exists an eigenvalue λjk ∈ spd (H . | λjk − λk ℏ2 Dz2 + µj f 2/(2+a) (z) | ≤ ℏ2 C .

Consequently, when k = 1, we have $ # tr((∂ 2 f (0))1/2 ) | ≤ ℏ2 C . | λj1 − µj + ℏ(µj )1/2 1/2 (2 + a)

(4.14)

(4.15)

4.3. Middle energies We can refine the preceding results when a ≥ 2, g ∈ C ∞ (Rm ) and f (∞) = ∞. We get then sharp localization near the µj ’s for much higher values of j’s. More precisely we prove: Theorem 4.3. . Assume the preceding properties, and consider j such that µj ≤ ℏ−2 ; then for any integer N , there exists a constant C depending only on N such * ℏ ) verifying that, for any k ≤ N , there exists an eigenvalue λjk ∈ spd (H . 2 2 2/(2+a) (y) | ≤ Cµj ℏ2 . (4.16) | λjk − λk ℏ Dy + µj f

Born-Oppenheimer-type Approximations Consequently, when k = 1, we have $ # tr((∂ 2 f (0))1/2 ) | λj1 − µj + ℏ(µj )1/2 | ≤ Cµj ℏ2 . 1/2 (2 + a)

411

(4.17)

4.4. An application We can apply the previous methods for studying Schr¨ odinger operators on L2 (Rds ) with d ≥ 2, P h = −h2 ∆ + V (s)

(4.18)

with a real and regular potential V (s) satisfying V ∈ C ∞ (Rd ; [0, +∞[) lim inf V (s) > 0

Γ = V

−1

(4.19)

|s|→∞

({0}) is a regular hypersurface.

Moreover we assume that Γ is connected and that there exist m ∈ N⋆ and C0 > 0 such that for any s verifying d(s, Γ) < C0−1 C0−1 d2m (s, Γ) ≤ V (s) ≤ C0 d2m (s, Γ)

(4.20)

(d(E, F ) denotes the euclidian distance between E and F ). We choose an orientation on Γ and a unit normal vector N (s) on each s ∈ Γ, and then, we can define the function on Γ,

2m 1 ∂ f (s) = N (s) V (s) , ∀ s ∈ Γ . (4.21) (2m)! ∂s Then by (4.19) and (4.20), f (s) > 0, ∀s ∈ Γ. Finally we assume that the function f achieves its minimum on Γ on a finite number of discrete points: Σ0 = f −1 ({η0 }) = {s1 , . . . , sℓ0 } ,

if η0 = min f (s) , s∈Γ

(4.22)

and the hessian of f at each point sj ∈ Σ0 is non degenerate. Hess (f )sj has d − 1 non negative eigenvalues ρ21 (sj ) ≤ · · · ≤ ρ2d−1 (sj ) , The eigenvalues

ρ2k (sj )

(ρj (sj ) > 0) .

do not depend on the choice of coordinates. We denote T r+ (Hess (f (sj ))) =

d−1

ρℓ (sj ) .

(4.23)

ℓ=1

We denote by (µj )j≥1 the increasing sequence of the eigenvalues of the operd2 ator − 2 + t2m on L2 (R). dt

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F. Truc

Theorem 4.4. Under the above assumptions, for any N ∈ N⋆ , there exist h0 ∈]0, 1] and C0 > 0 such that, if µj ≪ h−4m/(m+1)(2m+3) , and if α ∈ Nd−1 and |α| ≤ N , then ∀sℓ ∈ Σ0 , ∃λhjℓα ∈ spd (P h ) s.t. + , h 1/(m+1) 1/2 µj + h1/(m+1) µj Aℓ (α) λjℓα − h2m/(m+1) η0 2+3/2m

≤ h2 µj

with

C0 ;

1 2αρ(sℓ ) + T r+ (Hess (f (sℓ ))) . m/(2m+2) η0 (m + 1)1/2 (αρ(sℓ ) = α1 ρ1 (sℓ ) + · · · + αd−1 ρd−1 (sℓ )). Aℓ (α) =

References [1] J. Avron, I. Herbst, B. Simon, Duke. Math. J. 45 (1978), 847–883. [2] Y. Colin de Verdière, L’asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys. 105 (1986), 327–335. [3] J.S. de Wet and Mandl, On the asymptotic distribution of eigenvalues, Proc. Roy. Soc. London Ser. 200 (1950), 572–580. [4] B. Helffer: Introduction to the semiclassical analysis for the Schrödinger operator and applications. Springer Lecture Notes in Math., n0 1336 (1988). [5] B. Helffer, D. Robert, Propriétés asymptotiques du spectre d’opérateurs pseudodifférentiels sur Rn . Comm. in P.D.E., 7(7), (1982), pp. 795–882. [6] B. Helffer, D. Robert, Comportement semi-classique du spectre des hamiltoniens quantiques hypoelliptiques. Annales ENS Pise IV, 9,(3), (1982), pp. 405–431. [7] B. Helffer, J. Sjoestrand, Multiple wells in the semi-classical limit. I. Comm. in P.D.E., 9, (4), (1984), pp. 337–408. [8] B. Helffer, J. Sjoestrand, Puits multiples en mecanique semi-classique. VI. Ann. Institut H. Poincaré, Phys. Theor., 46,(4), (1987), pp. 353–372. [9] L. H¨ ormander, The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math., 32, (1979), pp. 359–443. [10] V. Ivrii, Microlocal analysis and precise spectral asymptotic. Springer, Berlin 1998. [11] B.M. Levitan, On the asymptotic behavior of Green’s function and its expansion in eigenvalues of Schr¨ odinger’s equation, Math. USSR-Sb.41 83 (1957), 439–458. [12] A.G. Kostjucenko, Asymptotic distribution of the eigenvalues of elliptic operators, Soviet Math. Dokl. 5 (1964), 1171–1175. [13] A. Martinez, Développement asymptotiques et effet tunnel dans l’approximation de Born-Oppenheimer. Ann. Inst. Henri Poincaré, Vol. 49,(3), (1989), pp. 239–257. [14] A. Morame, F. Truc, Semiclassical Eigenvalue Asymptotics for a Schrödinger Operator with Degenerate Potential. Asymptotic Anal., 22(1), (2000), pp. 39–49. [15] A. Morame, F. Truc, Accuracy on eigenvalues for a Schrödinger Operator with a Degenerate Potential in the semi-classical limit. (Submitted).


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[16] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Academic press, New York, 1978. [17] D. Robert, Comportement asymptotique des valeurs propres d’opérateurs du type de Schr¨ odinger ` a potentiel dégénéré, J. Math. Pures Appl. 61 (1982), 275–300. [18] G.V. Rosenbljum, Asymptotics of the eigenvalues of the Schrödinger operator, Math. USSR-Sb. 22(3) (1974), 349–371. [Shu] M.A. Shubin, Pseudodifferential operators and spectral theory. Springer-Verlag, Berlin 1987. [19] B. Simon, Non-classical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), 84–98. [20] B. Simon: Semi-classical analysis of low lying eigenvalues I. Ann. Inst. H. Poincaré, 38, (1983), pp. 295–307. [21] M.Z. Solomyak, Asymptotics of the spectrum of the Schrödinger operator with nonregular homogeneous potential, Math. USSR-Sb. 55(1) (1986), 19–37. [22] E.C. Titchmarsh, On the asymptotic distribution of eigenvalues, Quart. J. Math. Oxford Ser.(2) 5 (1954), 228–240. [23] F. Truc, Semi-classical asymptotics for magnetic bottles, Asympt. Anal. 15 (1997), 385–395. Fran¸coise Truc Université de Grenoble I Institut Fourier UMR 5582 CNRS-UJF B.P. 74 F-38402 St Martin d’Hères Cedex, FRANCE e-mail: [email protected]


The Ornstein-Uhlenbeck Semigroup in Bounded and Exterior Lipschitz Domains Ian Wood Abstract. We consider bounded Lipschitz domains Ω in Rn . It is shown that the Dirichlet-Laplacian generates an analytic C0 -semigroup on Lp (Ω) for p in an interval around 2 and that the corresponding Cauchy problem has the maximal Lq -regularity property. We then prove that for bounded or exterior Lipschitz domains Ornstein-Uhlenbeck operators A generate C0 -semigroups in the same p-interval. The method also allows to determine the domain D(A) of A and, if Ω satisfies an outer ball condition, allows to show Lp -Lq -smoothing properties of the semigroups. Mathematics Subject Classification (2000). 47D06, 47D07. Keywords. Ornstein-Uhlenbeck operator, Lipschitz domain, exterior domain, semigroup, maximal Lp -regularity.

1. Introduction We shall discuss the following problem. Let M be a real-valued constant coefficient n × n-matrix and let K be a compact set in Rn with Lipschitz boundary1 . Let Ω = Rn \ K be the exterior domain. On this domain we consider the following equation. ⎧ ⎨ ∂t u(x, t) = ∆u(x, t) + M x · ∇u(x, t) in Ω × R+ , u(x, t) = 0 on ∂Ω × R+ , (1.1) ⎩ u(x, 0) = u0 (x) for x ∈ Ω. Operators of the form ∆ + M x · ∇ are called Ornstein-Uhlenbeck operators.2

1 I.e., every point on the boundary of Ω has a neighbourhood U such that, after an affine change of coordinates, ∂Ω ∩ U is described by the equation xn = ϕ(x1 , . . . , xn−1 ) where ϕ is a Lipschitz continuous function and Ω ∩ U = {x ∈ U : xn > ϕ(x1 , . . . , xn−1 )} 2 Note that x is actually the space variable, so the notation Δ + M x · ∇ is formally bad notation, however it is the standard notation and we will use it in the following.

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In recent years, many authors have considered Ornstein-Uhlenbeck operators (see [4], [7], [13], [21], [23], [24]), either from the point of view of analysis or stochastics. All these articles deal with the realization of these operators in certain function spaces over the whole space Rn . One setting where Ornstein-Uhlenbeck operators arise naturally is in transforming parabolic equations such as the heat equation or the Navier-Stokes equations from rotating domains to a fixed domain Ω (see [16]). Therefore, there is considerable interest in such operators defined in exterior domains of Rn , see, e.g., [16], [15], [11]. In bounded domains, Ornstein-Uhlenbeck operators can be considered as relatively bounded perturbations of the Laplacian. If the domain is not bounded however, the term M x is unbounded which makes the problem much more difficult to consider. Before looking at Ornstein-Uhlenbeck operators, we need to study the Dirichlet-Laplacian in bounded Lipschitz domains. Consider Laplace’s equation ∆u = f in Ω,

u = 0 on ∂Ω,

and the heat equation u′ − ∆u = f in Ω,

u = 0 on ∂Ω,

u(0) = 0 in Ω,

where Ω ⊆ R is a bounded Lipschitz domain. The usual elliptic regularity results no longer hold for the Laplacian in Lipschitz domains (the domain of the operator is no longer contained in the Sobolev space W 2,p (Ω)) and it becomes a difficult problem to determine the domain of the operator. See [29] and the references therein for information on elliptic and parabolic problems in non-smooth domains. The aim of this paper is to collect the results on the Laplacian in Lipschitz domains and their application to Ornstein-Uhlenbeck operators in one place. The results on the Laplacian have previously appeared in [29] and those on the OrnsteinUhlenbeck operator in [12] in a very shortened form. The structure of the paper is the following. In the next section we consider the Dirichlet-Laplacian in Lipschitz domains. We first state known results for the elliptic problem, then we combine these with semigroup theory to show that the Dirichlet-Laplacian generates a C0 -semigroup in Lp (Ω) for a certain range of p. We then study some of the properties of the semigroup such as analyticity and positivity which finally lead to a maximal Lq -regularity result. In the last section we study Ornstein-Uhlenbeck operators. First, we look at the well-known Rn -case. We then study the first order drift term separately before looking at Ornstein-Uhlenbeck operators in bounded Lipschitz domains in Section 3.3. Using a cut-off procedure previously used by Hishida ([16], [17] and [18]) we combine the results in Rn and on bounded domains to obtain results for exterior domains in Section 3.4. Finally, Section 3.5 deals with Lipschitz domains satisfying a uniform outer ball condition which gives additional regularity of the solution when p ≤ 2. This allows us to show Lp -Lq -smoothing estimates which are useful for the study of semilinear equations. n

The Ornstein-Uhlenbeck Semigroup in Lipschitz Domains

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2. The Laplacian in Lipschitz domains 2.1. Laplace’s equation We first state the main result due to Jerison and Kenig on solutions to the Dirichlet problem in bounded Lipschitz domains. The spaces Lpα (Ω) denote the Bessel potential spaces of order α and exponent p over the domain Ω (cf. [19] for the definition and some properties). We remark that for Lipschitz domains Ω and k ∈ N, we have Lpk (Ω) = W k,p (Ω). Theorem 2.1. (Jerison, Kenig [19, Theorem 1.1]). Let Ω be a bounded Lipschitz domain in Rn , n ≥ 3. There exists ε ∈ (0, 1], depending only on the Lipschitz character3 of Ω such that for every f ∈ Lpα−2 (Rn ) there is a unique solution u ∈ Lpα (Ω) to the inhomogeneous Dirichlet problem ∆u = f in Ω, (2.1) u = 0 on ∂Ω, provided one of the following holds (cf. Figure 1):

½

½

« ½

¾

Figure 1. (α, 1/p)-region in which theorem holds 1 3 −1−ε 0 depends only on the Lipschitz character of Ω or • n ≥ 2 and suppose additionally that Ω satisfies a uniform outer ball condition and 1 < p < ∞. p Then the operator ∆w p generates a C0 -semigroup of contractions on L (Ω).


421

Proof. We start with the case n ≥ 3. Again we verify that the range condition p of the Lumer-Phillips-Theorem is satisfied. For u ∈ D(∆w p ), we have ∆u ∈ L (Ω), so ∆u ∈ Lpα−2 (Ω) for any α ≤ 2. By Theorem 2.1, there exists δ > 0 such that whenever 1 < p < 3 + δ, we can find α ∈ [1, 2] and a unique v ∈ Lpα (Ω) such that ∆v = ∆u in Ω, v=0 on ∂Ω, and vα,p ≤ C ∆uα−2,p . Then u−v is harmonic and by the Maximum Principle we have u = v. Thus u1,p ≤ uα,p ≤ C ∆uα−2,p ≤ C ∆up . p Therefore, : D(∆w p ) → L (Ω) is an injective mapping for our range of exponents p. A similar argument shows that it is also surjective. Thus under our assumptions we have 0 ∈ ρ(∆w p ) for any 1 < p < 3 + δ. Moreover, by Lemma 2.8, ∆w p is dissipative for p ≥ 2. This proves the theorem for 2 ≤ p < 3 + δ. For the case when (3 + δ)′ < p < 2 we consider the dual operator. Since ∆w p′ is ′ p m-dissipative, its dual operator ∆w p′ is m-dissipative in L (Ω) (cf. [6, Proposition w′ w w′ 3.10]). We now claim that ∆w p ⊆ ∆p′ , i.e., D(∆p ) ⊆ D(∆p′ ) and both operators ∞ w w coincide on D(∆p ). To see this, let v ∈ D(∆p ), u ∈ D(∆w p′ ), (vn ) ⊆ Cc (Ω) and ′ (un ) ⊆ Cc∞ (Ω) such that vn → v in W 1,p (Ω) and un → u in W 1,p (Ω). Then

∆w p

∆u, v = =

lim ∆u, vn = − lim ∇u, ∇vn

n→∞

n→∞

−∇u, ∇v = − lim ∇un , ∇v = lim un , ∆v = u, ∆v , n→∞

n→∞

′ w′ w so v ∈ D(∆w p′ ) and ∆p′ v = ∆v as claimed. Therefore ∆p is contained in a ′ dissipative operator and hence is itself dissipative for (3 + δ) < p ≤ 2. Moreover, as we have seen above, for these p the range condition is satisfied. Using the Lumer-Phillips Theorem, this completes the proof for n ≥ 3. For n = 2, we merely replace Theorem 2.1 by [19, Theorem 1.3], while in the case of domains satisfying a uniform outer ball condition we use Theorem 2.3 and argue in the same way obtaining the larger range of exponents p.

Remark 2.14. δ is given by 3ε/(1 − ε) where ε is the constant given in Theorem 2.1. Therefore, (3 + δ)′ = 3/(2 + ε). Corollary 2.15. If Ω ⊆ Rn , n ≥ 2, is a bounded Lipschitz domain satisfying a s uniform outer ball condition and 1 < p ≤ 2, then we have ∆w p = ∆p . w s w s Proof. Obviously, ∆sp ⊆ ∆w p . Since 0 ∈ ρ(∆p ) ∩ ρ(∆p ), we have ∆p = ∆p .

Remark 2.16. To make the statements more concise, in the following we will often only refer to the semigroup generated by the weak Dirichlet-Laplacian, recalling that whenever the strong Dirichlet-Laplacian is a generator, it coincides with the weak Dirichlet-Laplacian.

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2.2.2. Properties of the semigroup. We first show that the semigroups generated on Lp (Ω) are consistent. Proposition 2.17. Let Ω and n be as in Theorem 2.13 and suppose p and q satisfy the assumptions on p given there. Then the semigroups Tp generated by ∆w p and p q Tq generated by ∆w q are consistent, i.e., if f ∈ L (Ω) ∩ L (Ω) then Tp (t)f = Tq (t)f for all t ≥ 0.

w Proof. W.l.o.g. assume p < q. Then, as Ω is bounded, D(∆w q ) ⊆ D(∆p ). Let ). Then T (t)f is the unique classical solution to f ∈ D(∆w q q

D(∆w q )

u′ − ∆u = 0, u(0) = f

(2.4)

with Tq (t)f ∈ for t ≥ 0. But then Tq (t)f ∈ D(∆w p ), so it must agree with w Tp (t)f , the unique classical solution to (2.4) for f ∈ D(∆w p ). Since D(∆q ) is dense in Lq (Ω), we get Tp (t)f = Tq (t)f for all f ∈ Lq (Ω). This now gives us some further interesting results.

Theorem 2.18. Let Ω, n and p be as in Theorem 2.13 and let T denote the semigroup p generated by ∆w p on L (Ω). Then the following holds. w w 1. The spectrum of ∆w p is independent of p, i.e., σ(∆p ) = σ(∆2 ) for all p in the range. 2. T satisfies a Gaussian estimate, i.e., there exist constants a ≥ 0, M, b > 0 such that |T (t)f | ≤ M eat G(bt)|f | for t ≥ 0 where G(t)f = kt ∗ f is the Gaussian semigroup and kt is the Gaussian kernel. 3. T is analytic. 4. T is positive, i.e., T (t)f ≥ 0 for all f ≥ 0, t ≥ 0. 5. T is of negative type, i.e., the growth bound ω(T ) of the semigroup satisfies ω(T ) < 0. Moreover, ω(T ) is independent of p. Proof. We start with 2: On L2 (Ω), ∆w 2 is identical to the Dirichlet-Laplacian defined via the form in [3, Section 1]. Since the semigroups generated on Lp (Ω) are consistent, both for the construction of the generator in [3] and for the semigroups generated by ∆w p , the semigroups must coincide for all cases. Then by [3, Section 1] the semigroup satisfies a Gaussian estimate. 3 now follows from the Stein Interpolation Theorem, while 1 follows from the Gaussian estimate by a result due to Kunstmann [20]. To see 4, we first deal with with real-valued functions using [6, Corollary 7.15]. In this case, Lp (Ω) is a Banach lattice, therefore it is sufficient to show that ∆w p is dispersive. In the case p ≥ 2, integration by parts as in the proof of Lemma 2.8 proves this. When dealing with complex-valued functions, the positivity of the semigroup on Lp (Ω), p ≥ 2, obviously follows from the real-valued case and the fact that the operator has real-valued coefficients. Now let p < 2. Recall that a semigroup is positive iff the resolvent is positive for sufficiently large λ > 0 (see, e.g., [6, Proposition 7.1]). Let f ∈ Lp (Ω), f ≥ 0. Then there exist fn ∈ L2 (Ω), fn ≥ 0 such that fn → f in Lp (Ω). If ∆w p generates a


423

C0 -semigroup on Lp (Ω), then from the standard resolvent estimate for generators w p we obtain that R(λ, ∆w p )fn → R(λ, ∆p )f in L (Ω). However, positivity of the semigroup on L2 (Ω) and consistency of the semigroups imply R(λ, ∆w p )fn ≥ 0 and therefore R(λ, ∆w p )f ≥ 0 almost everywhere. This proves positivity also for the case p < 2. Finally, to prove 5, we use a result due to Weis (cf. [2, Theorem 5.3.6]) which states that for generators A of positive semigroups T on Lp (Ω), ω(T ) coincides w with the spectral bound s(A). By 1, we already know that σ(∆w p ) = σ(∆2 ), in particular equality holds for the spectral bound. It therefore remains to examine the case p = 2. Using Poincaré’s inequality, for u ∈ D(∆w 2 ), u = 0, we have 2

2

∆u, u = − ∇u2 ≤ −C u2 < 0. Then by [26, Theorem 1.3.9], we get that the numerical range and therefore the spectrum of ∆w 2 lie in the half-plane {z ∈ C : Re z ≤ −C}, in particular s(∆w 2 ) ≤ −C. 2.2.3. Maximal Lq -regularity. In the previous section we have proven generator results for the Dirichlet-Laplacian and gathered various properties of the semigroups and their generators. We can now exploit these results to show the maximal regularity property for the Laplacian. Theorem 2.19. Let Ω, n and p be as in Theorem 2.13. Then the weak DirichletLaplacian has the maximal regularity property, i.e., for 1 < q < ∞ and for every f ∈ Lq (R+ , Lp (Ω)) there exists a unique solution to ⎧ ′ ⎨ u (t) − ∆u(t) = f (t) for t ∈ R+ , u(t, x) = 0 on R+ × ∂Ω, (2.5) ⎩ u(0) = 0. 1,q (R+ , Lp (Ω)) and satisfies the estimate The solution u lies in Lq (R+ , D(∆w p ))∩W

uLq (R+ ,Lp (Ω)) + u′ Lq (R+ ,Lp (Ω)) + ∆uLq (R+ ,Lp (Ω)) ≤ C f Lq (R+ ,Lp (Ω)) . Proof. We have shown that the semigroup generated by ∆w p is contractive (Theorem 2.13), analytic and positive (Theorem 2.18) on Lp (Ω). Maximal regularity now follows from a result by Weis ([28, Corollary 4d]). Moreover, as the generated semigroup is of negative type (Theorem 2.18), we get u ∈ Lq (R+ , Lp (Ω)) (cf. [8]) which leads to the desired estimate using the Closed Graph Theorem. Of course, whenever the weak and the strong Laplacian coincide, this also yields maximal regularity for the strong Laplacian. Because of the importance of the result, in particular the better estimate (2.6), we state it here separately. Theorem 2.20. Let 1 < p ≤ 2, and Ω ⊆ Rn , n ≥ 2, be a bounded Lipschitz domain satisfying a uniform outer ball condition. Then the strong Dirichlet-Laplacian has the maximal Lq -regularity property in Lp (Ω), i.e., for 1 < q < ∞ and for every

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f ∈ Lq (R+ , Lp (Ω)) there exists a unique solution to (2.5). The solution u lies in Lq (R+ , W 2,p (Ω) ∩ W01,p (Ω)) ∩ W 1,q (R+ , Lp (Ω)) and satisfies the estimate uLq (R+ ,W 2,p (Ω)) + u′ Lq (R+ ,Lp (Ω)) ≤ C f Lq (R+ ,Lp (Ω)) .

(2.6)

Remark 2.21. The same results can be proven for exterior Lipschitz domains and exterior Lipschitz domains satisfying a uniform outer ball condition. To do this, construct solutions to the resolvent problem in the exterior domain by combining the solutions in bounded domains and the whole space Rn as is done in the next section for Ornstein-Uhlenbeck operators. Together with dissipativity of the operator this proves generation of a contractive C0 -semigroup via the Lumer-Phillips Theorem. Moreover, we see from the construction that the resolvent operators are consistent on the Lp -spaces, hence the generated semigroups are consistent and they coincide with the positive analytic semigroups gained by the form method in [3]. Invoking Weis’ Theorem once more, we obtain maximal Lq -regularity over Lp (Ω) for the Dirichlet-Laplacian in exterior Lipschitz domains with the same restrictions on the dimension, the domain and the exponent p as in the case of bounded Lipschitz domains.

3. The Ornstein-Uhlenbeck operator in Lipschitz domains 3.1. Ornstein-Uhlenbeck operators on Lp (Rn ) We define the following operator. For 1 < p < ∞ and M ∈ Rn×n , let / ARn u(x) := ∆u(x) + M x · ∇u(x), x ∈ Rn , D(ARn ) := {u ∈ W 2,p (Rn ) : M x · ∇u ∈ Lp (Rn )}.

(3.1)

We gather the main results on the Ornstein-Uhlenbeck operator ARn in a theorem. Theorem 3.1. Let 1 0, (4π)n/2 (det Qt )1/2 Rn

t T with Qt := 0 esM esM ds for t > 0. Moreover, there exists λ0 ∈ R such that for λ > λ0 the unique solution of the resolvent problem (λ − ARn )u = f satisfies uW 2,p (Rn ) + M x · ∇uLp (Rn ) ≤ Cλ f Lp (Rn )

(3.2)

for some constant Cλ depending on λ, and there exist constants C, ω independent of λ such that the solution to the resolvent problem satisfies the gradient estimate C ∇uLp (Rn ) ≤ (3.3) 1 f Lp (Rn ) . |λ − ω| 2 Furthermore, if 1 0, f ∈ Lp (Rn ), (3.4) p q

The Ornstein-Uhlenbeck Semigroup in Lipschitz Domains and for the gradient we have the estimate ; tA n ; ;∇e R f ; ≤ Ct− 21 − n2 ( p1 − q1 ) eωt f , t > 0, f ∈ Lp (Rn ). p q

425

(3.5)

Proof. By [27, Theorem 2.4], we obtain that ARn generates a positive C0 -semigroup on Lp (Rn ) and the explicit representation can be found in [23]. The estimate (3.2) then follows from the closed graph theorem. For the gradient estimate (3.3), using [15, Proposition 3.4], we see that the semigroup generated by ARn satisfies the estimate ; tA n ; ;∇e R f ; ≤ C1 eωt f (3.6) p p t2 for some ω ∈ R. For λ > λ0 , the resolvent is given by the Laplace transform of the semigroup (cf. [2]) and we get ; ∞ ; ∞ ; ; 1 e−λt ∇etARn f dt; t− 2 e(ω−λ)t dt f p ∇R(λ, ARn )f p = ; ; ; ≤ C 0

≤

C

p

Γ( 12 )

1

(λ − ω) 2

0

f p .

Finally, (3.4) and (3.5) follow from [15, Proposition 3.4 and Lemma 3.5].

3.2. The drift operator We have already studied the Laplacian in Lipschitz domains. We now take a closer look at the drift term before combining the results to study the Ornstein-Uhlenbeck operator. We first introduce the drift operator BΩ on a domain Ω ⊆ Rn : / BΩ u(x) := M x · ∇u(x), x ∈ Ω, (3.7) D(BΩ ) := {u ∈ W 1,p (Ω) : M x · ∇u ∈ Lp (Ω)}. Note that the condition M x · ∇u ∈ Lp (Ω) is trivially satisfied in bounded domains for u ∈ W 1,p (Ω). Moreover, in this case the drift term is a relatively bounded perturbation of the Laplacian: Proposition 3.2. Let D be a bounded Lipschitz domain. Then the drift operator p BD on D is relatively bounded by the weak Dirichlet-Laplacian ∆w p in L (D) for (3 + ε)′ 0 depends only on the Lipschitz constant of D. The relative bound is given by zero. w Proof. Obviously we have that D(BD ) ⊇ D(∆w p ). Let u ∈ D(∆p ). By Theorem 2.1, we can find δ > 0 such that

u1+δ,p ≤ C ∆up

for

(3 + ε)′ 0,

(3 + ε)′ < p < 3 + ε.

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Set Θ = δ/(1 + δ) ∈ (0, 1). Then, using the complex interpolation method (cf. [5]) and Jensen’s inequality, we obtain the estimate

Θ .1−Θ C (λ − ∆)up u1,p ≤ C ∆up λ ≤

=

Cλ−Θ ((1 − Θ) ∆up + Θ (λ − ∆)up ) Cλ−Θ ∆up + Cλ1−Θ up

As we can choose λ > 0 arbitrarily large, this concludes the proof.

The next step is to prove dissipativity for the drift operator. To do this, we need to integrate by parts. The next result shows that this is possible in our situation. Proposition 3.3. Let Ω be a Lipschitz domain in Rn , u ∈ W01,p (Ω), φ ∈ W 1,∞ (Rn ) and 1 < p < ∞. Then we have p − 2 p−2 φ u|u|p−2 ∂i u = − φ |u|p−2 u∂i u− φ |u| (u∂i u+u∂i u)− (∂i φ) |u|p . 2 Ω Ω Ω Ω (3.8) Proof. The proof is a simpler version of the proof of Theorem 3.1 in [25], as we only want to integrate by parts once. Proposition 3.4. Let Ω ⊆ Rn be a Lipschitz domain and let the drift operator BΩ be defined as in (3.7). Then BΩ + tr pM is dissipative in Lp (Ω) for all 1 < p < ∞. p

Proof. We need to show that Re BΩ u, u∗ ≤ − tr pM up where u∗ is as in Lemma 2.8. When Ω is unbounded, the term M x is unbounded which causes some problems. Therefore, we need to introduce a cut-off function. Let ϕ ∈ Cc∞ (Rn ), 0 ≤ ϕ ≤ 1 such that ϕ ≡ 1 on the unit ball B1 and supp ϕ ⊆ B2 . Let ϕR (x) := ϕ(x/R). Then |∇ϕR (x)| ≤ C/R and ∇ϕR converges pointwise to 0, while ϕR converges pointwise to 1. We have n M x · ∇u |u|p−2 u = Re mij xj (∂i u) |u|p−2 u Re BΩ u, u∗ = Re Ω

=

Re

n

i,j=1

lim

R→∞

Ω

i,j=1

Ω

ϕR mij xj (∂i u) |u|p−2 u,

where the last equality holds by Lebesgue’s Dominated Convergence Theorem. Now we apply Proposition 3.3 with φ = ϕR mij xj to obtain ϕR mij xj (∂i u) |u|p−2 u = − ϕR mij xj |u|p−2 u∂i u Ω Ω p − 2 p−2 − ϕR mij xj |u| (u∂i u + u∂i u) 2 Ω − ((∂i ϕR ) mij xj + ϕR δij mij ) |u|p . Ω


427

Applying the Dominated Convergence Theorem and collecting our results, we have n n Re BΩ u, u∗ = −Re mij xj |u|p−2 u∂i u − Re δij mij |u|p −Re

i,j=1 Ω n i,j=1

i,j=1

Ω

p − 2 p−2 |u| (u∂i u + u∂i u) mij xj 2 Ω

= −(p − 1) Re BΩ u, u∗ − tr M upp . This yields Re BΩ u, u∗ = −

tr M upp . p

Therefore, the operator BΩ − λ is dissipative for λ ≥ −tr M/p.

3.3. Ornstein-Uhlenbeck operators on bounded Lipschitz domains Let D be a bounded Lipschitz domain in Rn , n ≥ 3. We study the operator / AD u(x) := ∆u(x) + M x · ∇u(x), x ∈ D, (3.9) D(AD ) := {u ∈ W01,p (D) : ∆u ∈ Lp (D)}. Our main result for bounded domains is now an easy consequence of perturbation results for C0 -semigroups. Proposition 3.5. On bounded Lipschitz domains D ⊆ Rn , the operator AD generates an analytic quasi-contractive5 semigroup T for (3 + ε)′ 0 depends on the Lipschitz constant of the domain D. The growth bound of the semigroup can be estimated by ω(T ) ≤ − tr pM . For Reλ > − tr pM , there exists a constant Cλ depending on λ, such that we have the estimate uLp (D) + ∆uLp (D) + M x · ∇uLp (D) ≤ Cλ f Lp (D)

(3.10)

for solutions of the resolvent problem (λ − AD )u = f . Furthermore, the solution u satisfies the estimate uLp (D) ≤

C f Lp (D) |λ + tr pM |

(3.11)

for λ ∈ − tr pM + Σϕ for some angle ϕ, where Σϕ := {λ ∈ C : | arg λ| < ϕ} \{0} and arg λ denotes the argument of λ. For λ > max − tr pM , 0 , there exists Θ > 0 such that C ∇uLp (D) ≤ Θ f Lp (D) . (3.12) λ 5 I.e.,

there exists α ∈ R such that e−αt T (t) t≥0 is a contractive semigroup

428

I. Wood

Proof. Since, by Theorem 2.18, the weak Dirichlet-Laplacian generates an analytic semigroup of contractions, Proposition 3.2 implies that AD generates an analytic C0 -semigroup. The fact that it is quasi-contractive and the growth bound estimate follow from the dissipativity of the operator BD + tr pM (Proposition 3.4) and the Lumer-Phillips-Theorem (cf. [2]). Estimate (3.10) is a consequence of the closed graph theorem, while (3.11) is the standard resolvent estimate for analytic semigroups. For (3.12), we use the representation of the resolvent as R(λ, AD ) = R(λ, ∆w p)

∞

n (BD R(λ, ∆w p )) ,

n=0

where λ > max − tr pM , 0 . Due to the relative boundedness, the Neumann series gives a bounded operator, and we only need to show the corresponding estimate for the Laplacian. Now, a similar calculation as in the proof of Proposition 3.2 shows that for u ∈ D(AD ) we have u1,p

≤ Cλ−Θ (λ − ∆)up + Cλ1−Θ up .

Again using the fact that the Dirichlet-Laplacian generates an analytic semigroup of contractions, we can estimate λ up ≤ C (λ − ∆)up to obtain u1,p ≤ Cλ−Θ (λ − ∆)up . Therefore, ∇R(λ, AD )L(Lp (D)) ≤ Cλ−Θ concluding the proof.

3.4. Ornstein-Uhlenbeck operators on exterior Lipschitz domains Let Ω = Rn \ K where K is a bounded Lipschitz domain contained in a ball BR of radius R. Again we define the Ornstein-Uhlenbeck operator / AΩ u(x) := ∆u(x) + M x · ∇u(x), x ∈ Ω, (3.13) D(AΩ ) := {u ∈ W01,p (Ω) : ∆u ∈ Lp (Ω) and M x · ∇u ∈ Lp (Ω)}. Using the Lumer-Phillips Theorem, we will see that AΩ is a closed operator and generates a C0 -semigroup on Lp (Ω) for (3 + ε)′ 0 and (3 + ε)′ < p < 3 + ε. We use the following notation. The bounded Lipschitz domain Ω ∩ BR+3 will be denoted by D. We further introduce a cut-off function ϕ ∈ C ∞ (Rn ) with c ϕ(x) = 0 for x ∈ BR+1 and ϕ(x) = 1 for x ∈ BR+2 . Given f ∈ Lp (Ω), denote its trivial extension to K by f0 and its restriction to D by f∗ . By u0 we denote the solution to the resolvent problem (λ − ARn )u = f0 in Rn given in Theorem 3.1. Let u∗ denote the solution to (λ − AD )u = f∗ in D given by Proposition 3.5. Note that u0 ∈ W 2,p (Rn ) ∩ {u ∈ Lp (Rn ) : M x · ∇u ∈ Lp (Rn )} u∗ ∈ W01,p (D) and ∆u∗ ∈ Lp (D).


429

For the solution of the resolvent problem (λ − AΩ )u = f we make the ansatz u = Θλ f := ϕu0 + (1 − ϕ)u∗ . The function u then satisfies ∆u = ϕ∆u0 + (1 − ϕ)∆u∗ + 2∇ϕ · (∇u0 − ∇u∗ ) + ∆ϕ(u0 − u∗ )

and

M x · ∇u = ϕ M x · ∇u0 + (1 − ϕ)M x · ∇u∗ + (M x · ∇ϕ)(u0 − u∗ ).

Due to the properties of u0 and u∗ , it is easy to check that u ∈ W01,p (Ω), ∆u ∈ Lp (Ω) and M x · ∇u ∈ Lp (Ω). Furthermore, by estimates (3.2) and (3.10), we have uLp (Ω) + ∆uLp (Ω) + M x · ∇uLp (Ω) ≤ Cλ,ϕ f Lp (Ω) .

(3.14)

u satisfies the following equation. λu − ∆u − M x · ∇u = where

ϕf0 + (1 − ϕ)f∗ − 2∇ϕ · (∇u0 − ∇u∗ )

−∆ϕ(u0 − u∗ ) − (M x · ∇ϕ)(u0 − u∗ ) = (I − Tλ )f,

Tλ f := 2∇ϕ · (∇u0 − ∇u∗ ) + ∆ϕ(u0 − u∗ ) + (M x · ∇ϕ)(u0 − u∗ ).

Our aim is now to invert I − Tλ continuously in L(Lp (Ω)). We need the gradient estimate (3.3) for u0 , and for u∗ we use the estimates (3.11) and (3.12). Noting that ∇ϕ is supported in a compact set, for sufficiently large λ > 0 this yields Tλ f p

≤ C ∇u0 p + C ∇u∗ p + C u0 p + C u∗ p ≤

C 1 C f p ≤ f p . f p + |λ|Θ |λ| 2

Therefore, for sufficiently large λ > 0, I − Tλ can be continuously inverted in L(Lp (Ω)) using the Neumann series. Then we can solve the resolvent problem (λ − AΩ )u = f by setting u = Θλ (I − Tλ )−1 f . Theorem 3.6. Let Ω be an exterior Lipschitz domain. Then for (3 + ε)′ 0 depends only on the Lipschitz constant of the domain Ω, the operator AΩ defined as in (3.13) generates a quasi-contractive C0 -semigroup T on Lp (Ω). The growth bound can be estimated by ω(T ) ≤ − tr pM . Proof. The preceding discussion shows that the range condition of the LumerPhillips-Theorem is satisfied. By Proposition 3.4, the operator BΩ + tr pM is dissipative. Using a smooth cut-off function and the Dominated Convergence Theorem, we can integrate by parts to see that for p ≥ 2, the weak Dirichlet-Laplacian ∆w p is dissipative on exterior Lipschitz domains. This proves the theorem for 2 ≤ p < 3 + ε. For (3 + ε)′ < p < 2 we have so far only shown dissipativity of the weak Laplacian for bounded Lipschitz domains. However, simply setting M = 0 in the preceding discussion, we know that ∆w p is m-dissipative in exterior Lipschitz domains for 2 ≤ p < 3 + ε. The same consideration as in the proof of Theorem 2.13 then proves tr M also for (3 + ε)′ < p < 2 in exterior dissipativity of ∆w p and therefore of AΩ + p Lipschitz domains.

430

I. Wood

3.5. Domains satisfying an outer ball condition We now want to consider equation (1.1) in domains Ω satisfying a uniform outer ball condition where we can show that the domain of the operator ∆ + M x · ∇ is contained in W 2,p (Ω). To do this we proceed in the same manner as for general Lipschitz domains using the added regularity of solutions to the heat equation we proved in Theorem 2.20 for these domains. 3.5.1. Bounded domains. Let D be a bounded Lipschitz domain satisfying a uniform outer ball condition. On D, we define the operator / AD u(x) = ∆u(x) + M x · ∇u(x), x ∈ D, (3.15) D(AD ) = W 2,p (D) ∩ W01,p (D). Proposition 3.7. The drift operator BD on D given by BD u(x) = M x · ∇u(x), x ∈ D,

D(BD ) = W 1,p (D)

is relatively bounded by ∆sp in Lp (D) for 1 0, ; ; M x · ∇up ≤ C ∇up ≤ C(ε ;∇2 u;p + C(ε) up ) ≤ ε ∆up + C(ε) up .

As in the case of an arbitrary bounded Lipschitz domain, our generation result for bounded Lipschitz domains satisfying a uniform outer ball condition is now a direct consequence of the proposition and results from the perturbation theory of generators of contractive and analytic C0 -semigroups. Corollary 3.8. For 1 − tr pM , there exists a constant Cλ depending on λ such that we have the estimate uW 2,p (D) + M x · ∇uLp (D) ≤ Cλ f Lp (D)

(3.16)

for solutions of the resolvent problem (λ − AD )u = f . We now show that we also get so-called Lp -Lq -smoothing estimates for the generated semigroups. This kind of estimates is useful when studying semilinear equations. For example, in [15], Lp -Lq -smoothing estimates for the solution of the Stokes equation with linearly growing initial data are used to obtain mild solutions for the corresponding Navier-Stokes equations.


431

Lemma 3.9. Let D be a bounded Lipschitz domain satisfying a uniform outer ball condition. Let 1 − min tr pM , tr qM and all f ∈ Lp (D), ; tA ; ;e D f ; ≤ Ct− n2 ( p1 − q1 ) eωt f , t > 0, (3.17) p q and

; tA ; ;∇e D f ; ≤ Ct− 12 − n2 ( p1 − q1 ) eωt f , t > 0. p q

(3.18)

Proof. The proof is standard and relies on analyticity of the semigroup and the Gagliardo-Nirenberg inequality. We include it here for completeness. We first note that the semigroups generated by AD on the spaces Lp (Ω), 1 < p ≤ 2, are consistent which can be shown similarly as in the proof of Proposition 2.17 for the Laplacian. By (3.16) we have ; 2 ; ;∇ u; ≤ C (λ − AD )u ≤ C(AD u + u ), s

s

s

s

for sufficiently large λ, any u ∈ D(AD ) and 1 < s ≤ 2. Now choose

such that q11 ≤ n1 + q1 and such that − min tr pM , tr qM < ω ) < ω.

q1 ∈ [p, q] ω )

Then we have that a := n2 ( q11 − 1q ) ≤ 12 . Using the Gagliardo-Nirenberg inequality (cf. [22, Section 1.4.8]) with j = 0, l = 2, we then get ; ; ; ; ; ; ; tA ; ;e D f ; ≤ C ;∇2 etAD f ;a ;etAD f ;1−a + C ;etAD f ; q1 q q1 q1 ; ;a ; ;1−a ; ; ≤ C ;AD etAD f ;q1 ;etAD f ;q1 + C ;etAD f ;q1 ; ; t ;a ; t ;1−a ; t t t t ; ; ; ; ; ; ≤ C ;AD e 2 AD e 2 AD f ; ;e 2 AD e 2 AD f ; + C ;e 2 AD e 2 AD f ; q1 q1 q1 ; ; ;a (1−a))ωt ; t ;1−a ; ωt ; t ) ωt ; t ; ; ; ; A A A −a a) D D D + Ce 2 ;e 2 f ; ≤ Ct e 2 ;e 2 f ; e 2 ;e 2 f ; q1 q1 q1 ; t ; 1 n 1 ω ) t ω ) t ; 2 AD ; − 2 (q −q) 2 2 1 e + Ce ) ;e f; ≤ (Ct q1 ; ; 1 1 ωt ; t ; −n ( − ) ≤ Ct 2 q1 q e 2 ;e 2 AD f ; q1

for t > 0 and f ∈ Lp (Ω). Now choose q2 ∈ [p, q1 ] such that q12 ≤ By a similar calculation we see that ; t ; ; tA ; n 1 1 ; ; ;e D f ; ≤ Ct− 2 ( q2 − q ) e 3ωt 4 ;e 4 AD f ; q

1 n

+

1 q1

≤

q2

1 Iterating this procedure we obtain qm such that q1m ≤ m n + q and ; ; m ; tA ; 1 t ; 2m ; − 1q ) (2 2−1)ωt m ;e D f ; ≤ Ct− n2 ( qm e ;e AD f ; . q qm

After finitely many steps, we reach p and then (3.17) obviously is satisfied.

2 n

+ 1q .

432

I. Wood For (3.18), we use the Gagliardo-Nirenberg inequality in the form ; ; ∇uq ≤ C(;∇2 u;q + uq1 )a u1−a q1 1

1 + q1 and a := n2 ( q11 − q1 ) + where we choose q1 ∈ [p, q] such that q11 ≤ 2n Proceeding similarly to above then leads to the desired gradient estimate.

1 2

≤ 43 .

3.5.2. Exterior domains. Let Ω be an exterior Lipschitz domain in Rn satisfying a uniform outer ball condition. By this we mean that the complement of Ω is a compact set and that Ω itself is a Lipschitz domain and satisfies a uniform outer ball condition. Let the operator AΩ be defined by / AΩ u(x) = ∆u(x) + M x · ∇u(x), x ∈ Ω, (3.19) D(AΩ ) = {u ∈ W 2,p (Ω) : M x · ∇u ∈ Lp (Ω)} ∩ W01,p (Ω).

Then, as by Corollary 2.11 the strong Dirichlet-Laplacian is dissipative in Lp (Ω) for 1 0 and 1 < p ≤ 2. Using the same notation and proceeding just as before, we note that now D is a bounded Lipschitz domain satisfying a uniform outer ball condition. Therefore, u∗ ∈ W 2,p (D) ∩ W01,p (D) and u := ϕu0 + (1 − ϕ)u∗ ∈ W 2,p (Ω) ∩ W01,p (Ω). Instead of (3.14), we have the stronger estimate uW 2,p (Ω) + M x · ∇uLp (Ω) ≤ Cλ,ϕ f Lp (Ω) .

We summarise the generator result of this section in the following theorem. Theorem 3.10. Let Ω be an exterior Lipschitz domain satisfying a uniform outer ball condition. Then for 1 < p ≤ 2, the operator AΩ generates a quasi-contractive C0 -semigroup T on Lp (Ω). The growth bound can be estimated by ω(T ) ≤ − tr pM . Finally, we can extend the Lp -Lq -smoothing estimates for the generated semigroups to exterior domains. In order to do this, we use a lemma on iterated convolutions proved in [11]: Lemma 3.11. Let X, Y be Banach spaces and let T : (0, ∞) → L(Y, X) and S : (0, ∞) → L(Y ) be strongly continuous functions. Assume that T (t)L(Y,X) ≤ C0 tα eωt ,

S(t)L(Y ) ≤ C0 tβ eωt ,

t > 0,

for some C0 , ω > 0 and α, β > −1. For f ∈ Y , set T0 (t)f := T (t)f and Tn (t)f :=

t 0

Tn−1 (t − s)S(s)f ds,

Then there exist C, ω ˜ > 0 such that

∞

n=0

n ∈ N, t > 0.

Tn (t)f X ≤ Ctα eω˜ t f Y ,

t > 0.

Just as in the case of bounded domains, the statement of the Lp -Lq -smoothing properties in exterior domains reads as follows:


433

Theorem 3.12. Let 1 < p ≤ q ≤ 2 and let Ω be an exterior Lipschitz domain satisfying a uniform outer ball condition. Denote the semigroup generated by AΩ by etAΩ . Then there exist constants C, ω such that for all f ∈ Lp (Ω), ; tA ; ;e Ω f ; ≤ Ct− n2 ( p1 − q1 ) eωt f , t > 0, (3.20) p q and

; tA ; ;∇e Ω f ; ≤ Ct− 12 − n2 ( p1 − q1 ) eωt f , t > 0. p q

(3.21)

Proof. The proof is the same as the proof in [12] for C 1,1 -domains and is included here for completeness. Recall from the construction in Section 3.4 that for sufficiently large λ, the resolvent of λ − AΩ has the form (λ − AΩ )−1 f = Θλ (I − Tλ )−1 f, f ∈ Lp (Ω)

with Θλ and Tλ as in Section 3.4. Let f0 denote the trivial extension of f to Rn and fD denote the restriction of f to D. Then the Laplace transforms of the strongly continuous functions U : [0, ∞) → L(Lp (Ω)) and V : [0, ∞) → L(Lp (Ω)) given by U (t)f

:=

V (t)f

:=

ϕetARn f0 + (1 − ϕ)etAD fD ,

−2∇ϕ · ∇(etARn f0 − etAD fD )

−[∆ϕ + (M x · ∇ϕ)](etARn f0 − etAD fD )

are given by Θλ and Tλ , respectively. By the estimates (3.4), (3.6), (3.17) and (3.18), there exist constants C, ω such that U (t)L(Lp (Ω)) ≤ Ceωt ,

1

V (t)L(Lp (Ω)) ≤ Ct− 2 eωt ,

t > 0.

For f ∈ Lp (Ω) we set T0 (t)f := U (t)f and define Tn (t)f :=

t 0

Tn−1 (t − s)V (s)f ds,

It then follows from Lemma 3.11 that, TΩ (t)f :=

n ∈ N, ∞

t > 0.

Tn (t)f is well defined for all

n=0

t > 0 and exponentially bounded. Thus, by Lebesgue’s theorem ∞ ∞ ∞ ∞ ˆ (λ)Vˆ (λ)n = (λ − AΩ )−1 e−λt Tn (t)dt = e−λt TΩ (t)dt = U 0

n=0

n=0 0

tAΩ

for λ large enough and hence TΩ (t) = e for t ≥ 0. Now, consider U as a mapping U : [0, ∞) → L(Lp (Ω), Lq (Ω)). Once more using the estimates (3.4), (3.6), (3.17) and (3.18), there exist constants C, ω such that n 1 1 U (t)L(Lp(Ω),Lq (Ω)) ≤ Ct− 2 ( p − q ) eωt , t > 0.

Then (3.20) follows as above by Lemma 3.11 if n2 ( p1 − q1 ) < 1. In the general case, choose q1 such that n2 ( p1 − q11 ) < 1 and proceed as above. Then repeat this procedure until q is reached.

434

I. Wood

Estimate (3.21) follows in a similar way by defining W (t)f := ∇U (t)f , replacing U (t) in the above proof by W (t) and using the estimates (3.5) and (3.18) on the gradient of the semigroup. Remark 3.13. Note that due to the representation of the semigroup using only the semigroups on the whole space Rn and on the bounded domain D which are consistent, the semigroups on exterior domains are also consistent.

References [1] R.A. Adams: Sobolev Spaces, Academic Press, New York, 1975. [2] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics 96, Birkh¨ auser, Basel 2001. [3] W. Arendt, P. Bénilan: Wiener regularity and heat semigroups on spaces of continuous functions. Progress in Nonlinear Diff. Eq. and their applications 35 (1999), 29–49. [4] D. Bakry: L’hypercontractivité et son utilisation en théorie des semigroupes. In: Lectures on probability theory (Saint-Flour, 1992). Lecture Notes in Math., vol. 1581, (1994) Springer, Berlin, 1–114. [5] J. Bergh, J. L¨ ofstr¨ om: Interpolation Spaces. Springer, Berlin 1976. [6] P. Clément, H.J.A.M. Heijmans et al.: One-Parameter Semigroups. CWI Monographs 5, North-Holland, Amsterdam (1987). [7] G. Da Prato, A. Lunardi: On the Ornstein-Uhlenbeck operator in spaces of continuous functions. J. Func. Anal. 131 (1995), no. 1, 94–114. [8] G. Dore: Lp -regularity for abstract differential equations. In: Functional Analysis and Related Topics, H. Komatsu (ed.), Lecture Notes in Math. 1540. Springer, 1993. [9] K.-J.Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Springer, 2000. [10] S.J. Fromm: Potential space estimates for Green potentials in convex domains. Proc. Amer. Math. Soc. 119, 1 (1993), 225–233. [11] M. Geissert, H. Heck, M. Hieber: Lp -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, (2006), 45–62. [12] M. Geissert, H. Heck, M. Hieber, I. Wood: The Ornstein-Uhlenbeck semigroup in exterior domains. Arch. Math., 85, 6 (2005), 554–562. [13] B. Goldys: On analyticity of Ornstein-Uhlenbeck semigroups. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10, 3 (1999), 131–140. [14] P. Grisvard: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. [15] M. Hieber, O. Sawada: The Navier-Stokes equations in Rn with linearly growing initial data. Arch. Rat. Mech. Anal., 175, 2 (2005), 269–285. [16] T. Hishida: An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. Arch. Rational Mech. Anal. 150 (1999), 307–348. [17] T. Hishida: The Stokes operator with rotation effect in exterior domains. Analysis 19 (1999), 51–67.


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[18] T. Hishida: L2 -theory for the operator Δ + (k × x) · ∇ in exterior domains. Nihonkai Math. J. 11 (2000), 103–135. [19] D. Jerison, C.Kenig: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Func. Anal. 130 (1995), 161–219. [20] P.C. Kunstmann: Heat kernel estimates and Lp spectral independence of elliptic operators. Bull. London Math. Soc. 31 (1999), 345–353. [21] A. Lunardi, V. Vespri: Generation of strongly continuous semigroups by elliptic operators with unbounded coefficients in Lp (Rn ). Rend. Instit. Mat. Univ. Trieste 28 (1996), no. suppl., 251–279. [22] V.G. Maz’ja: Sobolev Spaces. Springer, Berlin, 1985. [23] G. Metafune: Lp -spectrum of Ornstein-Uhlenbeck operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 97–124. [24] G. Metafune, J. Pr¨ uss, A. Rhandi, R. Schnaubelt: The domain of the OrnsteinUhlenbeck operator on an Lp -space with invariant measure. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2002), 471–485. [25] G. Metafune, C. Spina: An integration by parts formula in Sobolev spaces.Ulmer Seminare u ¨ber Funktionalanalysis und Differentialgleichungen (2004). [26] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, 1983. [27] J. Pr¨ uss, A. Rhandi, R. Schnaubelt: The domain of elliptic operators on Lp (Rd ) with unbounded drift coefficients. Preprint. [28] L. Weis: A new approach to maximal Lp -regularity. In: Evolution Equations and Appl. Physical Life Sciences, G. Lumer, L. Weis (eds.), Lect. Notes in Pure & Appl. Math., Vol. 215 Marcel Dekker, New York (2001), 195–214. [29] I. Wood: Maximal Lp -regularity for the Laplacian on Lipschitz domains, Math. Z. (4) 255 (2007), 855–875. Ian Wood Institute of Mathematical and Physical Sciences Aberystwyth University Penglais Aberystwyth SY23 3BZ, UK e-mail: [email protected]


List of Participants

Organizers

Participants

Jan Janas Institute of Mathematics PAN Branch in Cracow Sw.Tomasz 30, 31-027, Cracow, Poland [email protected]

Vadym Adamyan Department of Theor. Physics Odessa National I.I. Mechnikov University vul. Dvoryanska 2, 65026 Odessa, Ukraine [email protected]

Pavel Kurasov Dept of Mathematics, LTH S¨ olvegatan 18, Box 118 221 00 Lund, Sweden [email protected]

Sergei A. Avdonin Department of Mathematics and Statistics University of Alaska Fairbanks Fairbanks, AK 99775-6660, USA [email protected]

Ari Laptev Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden [email protected] Sergey Naboko Department of Mathematical Physics Institute of Physics NIIF, St. Petersburg State University ul.Ulianovskaya 1, St. Petergoff 198504 St. Petersburg, Russia [email protected] Gunter Stolz Department of Mathematics, CH 542 University of Alabama at Birmingham 1300 University Blvd Birmingham, AL 35294-1170, USA [email protected]

Ingrid Beltit¸˘a Institute of Mathematics “Simion Stoilow” of the Romanian Academy Calea Grivitei 21, PO Box 1–764 RO 014700, Bucharest, Romania [email protected] Jussi Behrndt Institut f¨ ur Mathematik MA 6-4, TU Berlin Strasse des 17. Juni 136 10623 Berlin, Germany [email protected] Christer Bennewitz Department of Mathematics Lund University, P.O. Box 118 221 00 Lund, Sweden [email protected]

438


Tanja Bergkvist Department of Mathematics Stockholm University 106 91 Stockholm, Sweden [email protected]

Evgeney V. Cheremnikh Institute of Mathematics 12 Bandera Str. 79013 Lviv, Ukraine [email protected]

Gregory Berkolaiko Dept of Mathematics Texas A&M University College Station, TX 77843-3368, USA [email protected]

Jacob S. Christiansen Caltech, Mathematics 253-37, Pasadena, CA 91125, USA [email protected]

Jan Boman Department of Mathematics Stockholm University 106 91 Stockholm, Sweden [email protected] Julius Borcea Department of Mathematics Stockholm University 106 91 Stockholm, Sweden [email protected] Denis I. Borisov Department of Theoretical Physics Nuclear Physics Institute 250 68 Rez, Czech Republic [email protected] Jonathan Breuer Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram The Hebrew University of Jerusalem Jerusalem, 91904, Israel [email protected] Malcolm B. Brown Computer Science, Cardiff University, Queen’s Buildings, 5 The Parade, Roath Cardiff CF24 3AA, UK [email protected] Vsevolod L. Chernyshev Fundamental Sciences Moscow State Technical University n.a. N.E. Bauman 2-nd Baumanskaya 5 105005 Moscow, Russia [email protected]

Monique Combescure IPNL, 4 rue Enrico Fermi F-69622 Villeurbanne , France [email protected] Rafael R. del Rio IIMAS-UNAM, Circuito Escolar Ciudad Universitaria 04510 Mexico D.F., Mexico [email protected] Nils Dencker Center for Mathematical Sciences Division of Mathematics (Faculty of Science) Lund University Box 118, 221 00 Lund, Sweden [email protected] Jan Derezi´ nski Dep. Math. Meth. in Physics, Warsaw University Hoza 74, 00-682 Warszawa, Poland [email protected] Joanne M. Dombrowski Department of Mathematics and Statistics Wright State University Dayton, Ohio 45434, USA [email protected] Tomas Ekholm Complexo Interdisciplinar da Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal [email protected]

List of Participants Mats Ehrnstr¨ om Centre for Mathematical Sciences Lund University P.O. Box 118, 221 00 Lund, Sweden [email protected] Domenico Finco Institut fur Angewandte Mathematik Abteilung Stochastik Wegelerstr. 6, D-53115 Bonn, Germany [email protected] Magnus Fontes Department of Mathematics (LTH) Lund University P.O. Box 118, 221 00 Lund, Sweden [email protected] Rupert Frank KTH Stockholm Department of Mathematics 100 44 Stockholm, Sweden [email protected] David F¨ arm Mathematics LTH Box 118, 221 00 Lund, Sweden [email protected] Clemens F¨ orster Institut f¨ ur Analysis Dynamik und Modellierung Fakult¨ at f¨ ur Mathematik und Physik Universit¨ at Stuttgart Pfaffenwaldring 57 70569 Stuttgart, Germany Clemens.Foerster @mathematik.uni-stuttgart.de Jeffrey S. Geronimo School of Mathematics, Georgia Tech Atlanta, Ga 30332, USA [email protected] Fritz Gesztesy Department of Mathematics University of Missouri Columbia, MO-65211, USA [email protected]

439

Daphne J. Gilbert Department of Pure and Applied Mathematics Dublin Institute of Technology Kevin Street, Dublin 8, Ireland [email protected] Daniel Grieser Institut f¨ ur Mathematik Carl von Ossietzky Universität Oldenburg D-26111 Oldenburg, Germany [email protected] Thomas Guhr Mathematical Physics, LTH Lund University Box 118, 221 00 Lund, Sweden [email protected] Anders Hansson KTH Matematik 100 44 Stockholm, Sweden [email protected] Anders Holst Mathematics LTH, Lund University P.O. Box 118, 221 00 Lund, Sweden [email protected] Alexei Iantchenko Malm¨ o H¨ ogskola, Teknik och Samh¨ alle 205 06 Malm¨ o, Sweden [email protected] Per-Anders Ivert Lund University, Centre for Mathematical Sciences P.O. Box 118, 221 00 Lund, Sweden [email protected] Yulia Karpeshina Department of Mathematics, UAB 1300 University blvd Birmingham, AL 35294-1170, USA [email protected]

440


Steffen Klassert Technische Universität Chemnitz Fakult¨ at f¨ ur Mathematik Reichenhainerstr. 41 09107 Chemnitz, Germany [email protected]

Tatiana Levitina Abteilung Partielle Differentialgleichungen Institut Computational Mathematics TU Braunschweig, Pockelsstr. 14 - Forum D-38106 Braunschweig, Germany [email protected]

Torbj¨ orn Kolsrud Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden [email protected]

Annemarie Luger Technische Universität Berlin Sekretariat MA 674 Strasse des 17. Juni 136 10623 Berlin, Germany [email protected]

Sylwia M. Kondej Department of Physics and Astronomy Institute of Physics ul. Szafrana 4a (room 428) 65-516 Zielona Gora, Poland [email protected]

Marco Marletta Cardiff School of Mathematics Senghennydd Road Cardiff CF24 4AG Wales, United Kingdom [email protected]

Evgeny Korotyaev Institut f¨ ur Mathematik Humboldt Universit¨ at zu Berlin Rudower Chaussee 25 12489 Berlin, Germany [email protected] Hynek Kovaˇr´ık Institute for Analysis Dynamics and Modelling Pfaffenwaldring 57 D-70569 Stuttgart, Germany [email protected] Stanislav Kupin CMI, Université de Provence 39 rue Joliot-Curie 13453 Marseille Cedex 13, France [email protected] Yoram Last Institute of Mathematics The Hebrew University Givat Ram, 91904 Jerusalem, Israel [email protected]

Asya A. Metelkina LAGA, Institut Galilée Université Paris 13 99 avenue Jean-Baptiste Clément 93430 Villetaneuse, France [email protected] Victor S. Mikhaylov University os Alaska 101 Chapman Hall Department of Mathematics Fairbanks, AK 99775, USA [email protected] Stanislav A. Molchanov Department of Mathematics and Statistics University of North Carolina at Charlotte Charlotte, NC 28223, USA [email protected] Marcin Moszy´ nski Wydzial Matematyki Informatyki i Mechaniki UW Instytut Matematyki Stosowanej i Mechaniki ul. Banacha 2 02-097 Warszawa, Poland [email protected]

List of Participants Wojciech Motyka Institute of Mathematics of the Polish Academy of Sciences ul. Sw. Tomasza 30 31-027 Krakow, Poland [email protected] Hagen Neidhardt WIAS-Berlin, Mohrenstr. 39 D-10117 Berlin, Germany [email protected] Johan Nilsson Mathematics LTH Box 118, 221 00 Lund, Sweden [email protected] Marlena Nowaczyk Mathematics LTH Box 118, 221 00 Lund, Sweden [email protected] Andrey Osipov 6 Parkovaya St., 29 “A”, flat 18 105043 Moscow, Russia [email protected] Jorgen S. Ostensson Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200B B3001 Leuven, Belgium [email protected] Elin Ottergren Department of Mathematics Roslagsv. 101 106 91 Stockholm, Sweden [email protected] Kateˇrina Oˇzanová Department of Mathematical Sciences Chalmers University of Technology Chalmers Tvargata 3 412 96 Goteborg, Sweden [email protected]

441

Konstantin Pankrashkin Institute of Mathematics Humboldt-University of Berlin Rudower Chaussee 25 12489 Berlin, Germany [email protected] Boris S. Pavlov Department of Mathematics University of Auckland private box 92019 Auckland, New Zealand [email protected] Tomas Persson Mathematics LTH Box 118, 221 00 Lund, Sweden [email protected] Irina Peterburgsky Department of Mathematics and Computer Science Suffolk University 41 Temple Street Boston, MA 02114, USA [email protected] Pelle Pettersson Department of Mathematics (LTH) Lund University P.O. Box 118, 221 00 Lund, Sweden [email protected] Victoria Rayskin Department of Mathematics PO Box 750561, 28725 Bremen, Germany [email protected] Roman V. Romanov Laboratory of Quantum Networks Institute for Physics Saint Petersburg State University 198504 St. Petersburg, Russia [email protected] Aida M. Sahmurova Istanbul University, Engineering Faculty Department of Environmental Engineering 34320 Avcilar/Istanbul, Turkey [email protected]

442


J¨ org Schmeling Mathematics LTH Box 118, 221 00 Lund, Sweden [email protected]

Jan Stochel Instytut Matematyki, ul. Reymonta 4 30-059 Krakow, Poland [email protected]

Achim Schroll Centre for Mathematical Sciences Box 118, 221 00 Lund, Sweden [email protected]

Jan E. Str¨ ang Abteilung Theoretische Physik Universit¨ at Ulm Albert-Einstein-Allee 11 89069 Ulm, Germany [email protected]

Veli B. Shakhmurov Istanbul University, Engineering Faculty Department of Electrical & Electronics Engineering Avcilar, 34320 Istanbul, Turkey [email protected] Luis O. Silva P. Departamento de Métodos Matem´ aticos y Numéricos IIMAS Univ. Nacional Aut´ onoma de México Apdo. Postal 20-726 México, D.F. 01000, México [email protected] Sergei Silvestrov Centre for Mathematical Sciences LTH Lund University Box 118, 221 00 Lund, Sweden [email protected] Anna-Maria Simbotin Centre for Mathematical Sciences Box 118, 221 00 Lund, Sweden [email protected] Barry Simon Mathematics 253-37, Caltech Pasadena, CA 91125, USA [email protected] Sergey A. Simonov Department of Mathematical Physics Institute of Physics St. Petersburg University Ulianovskaia 1 198504, St. Petergoff St. Petersburg, Russia sergey [email protected]

Vladimir V. Sukhanov Dep. of Mathematical Physics St. Petersburg Univ. Ulyanovskaya 1 198504 St. Petersburg, Russia [email protected] Alexander Teplyaev Department of Mathematics University of Connecticut 196 Auditorium Road U-3009 Storrs, CT 06269-3009, USA [email protected] Gerald Teschl Faculty of Mathematics Nordbergsr. 15 A-1090 Vienna, Austria [email protected] Alexey S. Tikhonov Dep. of Mathematics 4 Yaltinskaya 95007 Simferopol, Ukraine [email protected] Fran¸coise Truc Department of Mathematics Institut Fourier 100 rue des maths 38402 St Martin d’Hères, France [email protected] Tatyana Turova Department of Mathematical Statistics Lund University Box 118, 221 00 Lund, Sweden [email protected]

List of Participants Thomas Vallier Centre for Mathematical Sciences Lund University Box 118, 221 00 Lund, Sweden [email protected] Olivier Verdier Mathematics LTH Box 118, 221 00 Lund, Sweden [email protected] Ivan Veselic’ TU Chemnitz, Fakult¨ at f¨ ur Mathematik Room 428 in Reichenheiner Str. 39 09107 Chemnitz, Germany ivan.veselic @mathematik.tu-chemnitz.de Javier A. Villamizar V. Department of Mathematics Simon Bolivar University Apdo 89000 Caracas 1080-A, Venezuela [email protected] Erik Wahlén Lund University Centre for Mathematical Sciences P.O. Box 118 S-221 00 Lund, Sweden [email protected] Kazuo Watanabe Department of Mathematics Gakushuin Univ. 1-5-1 Mejiro, Toshima-ku Tokyo 171-8588, Japan [email protected]

Ricardo Weder IIMAS-UNAM Apartado Postal 20-726 Mexico DF 01000, Mexico [email protected] Rudi Weikard Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170, USA [email protected] Jens Wittsten Centre for Mathematical Sciences Lund University P.O. Box 118, 221 00 Lund, Sweden [email protected] Ian G. Wood Cardiff School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG UK [email protected] Peter Yuditskii Mathematical Department of Bar Ilan University 52900 Ramat Gan, Israel [email protected] Alessandro Zampini SiSSA, via Beirut 4 34014 Trieste, Italy [email protected]

443

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